From Pseudorandom Walk to Pseudo-Brownian Motion: First Exit Time from a One-Sided or a Two-Sided Interval

Abstract

Let 𝑁 be a positive integer, 𝑐 a positive constant and (𝜉_𝑛)_𝑛≥1 be a sequence of independent identically distributed pseudorandom variables. We assume that the 𝜉_𝑛’s take their values in the discrete set {−𝑁,−𝑁+1,…,𝑁−1,𝑁} and that their common pseudodistribution is characterized by the (positive or negative) real numbers ℙ{𝜉_𝑛=𝑘}=𝛿_𝑘0+(−1)^𝑘−1𝑐^2𝑁_𝑘+𝑁 for any 𝑘∈{−𝑁,−𝑁+1,…,𝑁−1,𝑁}. Let us finally introduce (𝑆_𝑛)_𝑛≥0 the associated pseudorandom walk defined on ℤ by 𝑆₀=0 and 𝑆_𝑛=∑^𝑛_𝑗=1𝜉_𝑗 for 𝑛≥1. In this paper, we exhibit some properties of (𝑆_𝑛)_𝑛≥0. In particular, we explicitly determine the pseudodistribution of the first overshooting time of a given threshold for (𝑆_𝑛)_𝑛≥0 as well as that of the first exit time from a bounded interval. Next, with an appropriate normalization, we pass from the pseudorandom walk to the pseudo-Brownian motion driven by the high-order heat-type equation 𝜕/𝜕𝑡=(−1)^𝑁−1𝑐𝜕^2𝑁/𝜕𝑥^2𝑁. We retrieve the corresponding pseudodistribution of the first overshooting time of a threshold for the pseudo-Brownian motion (Lachal, 2007). In the same way, we get the pseudodistribution of the first exit time from a bounded interval for the pseudo-Brownian motion which is a new result for this pseudoprocess.

1. Introduction

Throughout the paper, we denote by ℤ the set of integers, by ℕ that of nonnegative integers, and by ℕ^∗ that of positive integers: ℤ={…,−1,0,1,…}, ℕ={0,1,2,…}, ℕ^∗={1,2,…}. More generally, for any set of numbers 𝐸, we set 𝐸^∗=𝐸⧵{0}.

Let 𝑁 be a positive integer, 𝑐 a positive constant, and set 𝜅_𝑁=(−1)^𝑁−1. Let (𝜉_𝑛)_𝑛∈ℕ^∗ be a sequence of independent identically distributed pseudorandom variables taking their values in the set of integers {−𝑁,−𝑁+1,…,−1,0,1,…,𝑁−1,𝑁}. By pseudorandom variable, we mean a measurable function defined on a space endowed with a signed measure with a total mass equaling the unity. We assume that the common pseudodistribution of the 𝜉_𝑛’s is characterized by the (positive or negative) real pseudo-probabilities 𝑝_𝑘=ℙ{𝜉_𝑛=𝑘} for any 𝑘∈{−𝑁,−𝑁+1,…,𝑁−1,𝑁}. The parameters 𝑝_𝑘 sum to the unity: ∑^𝑁_𝑘=−𝑁𝑝_𝑘=1.

Now, let us introduce (𝑆_𝑛)_𝑛∈ℕ the associated pseudorandom walk defined on ℤ by 𝑆₀=0 and 𝑆_𝑛=∑^𝑛_𝑗=1𝜉_𝑗 for 𝑛∈ℕ^∗. The infinitesimal generator associated with (𝑆_𝑛)_𝑛∈ℕ is defined, for any function 𝑓 defined on ℤ, as 𝒢_𝑆𝑓𝑗=𝔼𝑓𝜉₁𝑗=+𝑗−𝑓_{𝑁𝑘=−𝑁}𝑝_𝑘𝑓𝑗+𝑘−𝑓𝑗,𝑗∈ℤ. (1) Here we consider the pseudorandom walk which admits the discrete 𝑁-iterated Laplacian as a generator infinitesimal. More precisely, by introducing the so-called discrete Laplacian Δ defined, for any function 𝑓 defined on ℤ, by Δ𝑓𝑗=𝑓𝑗+1−2𝑓𝑗+𝑓𝑗−1,𝑗∈ℤ, (2) the discrete 𝑁-iterated Laplacian is the operator Δ^𝑁=Δ∘⋯∘Δ_𝑁times given by Δ^𝑁𝑓𝑗=^𝑁_𝑘=−𝑁(−1)^𝑘+𝑁𝑓2𝑁𝑘+𝑁𝑗+𝑘,𝑗∈ℤ. (3) We then choose the 𝑝_𝑘’s such that 𝒢_𝑆=𝜅_𝑁𝑐Δ^𝑁 which yields, by identification, for any 𝑘∈{−𝑁,−𝑁+1,…,−1,1,…,𝑁−1,𝑁}, 𝑝_𝑘=𝑝_−𝑘=(−1)^𝑘−1𝑐2𝑁𝑘+𝑁,𝑝₀𝑁=1−𝑐.2𝑁 (4)

When 𝑁=1, (𝑆_𝑛)_𝑛∈ℕ is the nearest neighbours pseudorandom walk with a possible stay at its current location; it is characterized by the numbers 𝑝₁=𝑝₋₁=𝑐 and 𝑝₀=1−2𝑐. Moreover, if 0<𝑐<1/2, then 𝑝₀>0; in this case, we are dealing with an ordinary symmetric random walk (with positive probabilities). If 𝑐=1/2, this is the classical symmetric random walk: 𝑝₁=𝑝₋₁=1/2 and 𝑝₀=0.

Actually, with the additional assumption that 𝑝_−𝑘=𝑝_𝑘 for any 𝑘∈{−𝑁,−𝑁+1,…,𝑁−1,𝑁} (i.e., the 𝜉_𝑛’s are symmetric, or the pseudorandom walk has no drift), the 𝑝_𝑘’s are the unique numbers such that 𝒢_𝑆𝑓𝑗=𝜅_𝑁𝑐𝑓^(2𝑁)𝑗+termswithhigherorderderivatives, (5)

where 𝑓 is an analytical extension of 𝑓 and 𝑓^(2𝑁) stands for the (2𝑁)th derivative of 𝑓∶𝑓^(2𝑁)(𝑥)=(d^2𝑁𝑓/d𝑥^2𝑁)(𝑥).

Our motivation for studying the pseudorandom walk associated with the parameters defined by (4) is that it is the discrete counterpart of the pseudo-Brownian motion as the classical random walk is for Brownian motion. Let us recall that pseudo-Brownian motion is the pseudo-Markov process (𝑋_𝑡)_𝑡≥0 with independent and stationary increments, associated with the signed heat-type kernel 𝑝(𝑡;𝑥) which is the elementary solution of the high-order heat-type equation 𝜕/𝜕𝑡=𝜅_𝑁𝑐𝜕^2𝑁/𝜕𝑥^2𝑁. The kernel 𝑝(𝑡;𝑥) is characterized by its Fourier transform: 𝔼e^{𝑖𝑢𝑋𝑡}=^+∞_−∞e^𝑖𝑢𝑥𝑝(𝑡;𝑥)d𝑥=e^{−𝑐𝑡𝑢2𝑁}. (6) The corresponding infinitesimal generator is given, for any 𝐶^2𝑁-function 𝑓, by 𝒢_𝑋𝑓(𝑥)=lim_ℎ→0+1ℎ𝔼𝑓𝑋_ℎ+𝑥−𝑓(𝑥)=𝜅_𝑁𝑐𝑓^(2𝑁)(𝑥). (7)

The reader can find extensive literature on pseudo-Brownian motion. For instance, let us quote the works of Beghin et al. [1–20] and the references therein.

We observe that (5) and (7) are closely related to the continuous 𝑁-iterated Laplacian d^2𝑁/d𝑥^2𝑁. For 𝑁=2, the operator Δ² is the two-Laplacian related to the famous biharmonic functions: in the discrete case, Δ²𝑓𝑗=𝑓𝑗+2−4𝑓𝑗+1+6𝑓𝑗−4𝑓𝑗−1+𝑓𝑗−2,𝑗∈ℤ, (8) and in the continuous case, Δ²𝑓(𝑥)=d⁴𝑓d𝑥⁴(𝑥),𝑥∈ℝ. (9)

In the discrete case, it has been considered by Sato [21] and Vanderbei [22].

The link between the pseudorandom walk and pseudo-Brownian motion is the following one: when normalizing the pseudorandom walk (𝑆_𝑛)_𝑛∈ℕ on a grid with small spatial step 𝜀^2𝑁 and temporal step 𝜀 (i.e., we construct the pseudoprocess (𝜀𝑆_{⌊𝑡/𝜀2𝑁⌋})_𝑡≥0 where ⌊⋅⌋ denotes the usual floor function), the limiting pseudoprocess as 𝜀→0⁺ is exactly the pseudo-Brownian motion.

Now, we consider the first overshooting time of a fixed single threshold 𝑎<0 or 𝑏>0 (𝑎,𝑏 being integers) for (𝑆_𝑛)_𝑛∈ℕ: 𝜎⁻_𝑎=min𝑛∈ℕ^∗∶𝑆_𝑛,𝜎≤𝑎_+𝑏=min𝑛∈ℕ^∗∶𝑆_𝑛≥𝑏 (10)

as well as the first exit time from a bounded interval (𝑎,𝑏): 𝜎_𝑎𝑏=min𝑛∈ℕ^∗∶𝑆_𝑛≤𝑎or𝑆_𝑛≥𝑏=min𝑛∈ℕ^∗∶𝑆_𝑛∉(𝑎,𝑏) (11)

with the usual convention that min∅=+∞. Hence, when 𝜎⁺_𝑏<+∞, 𝑆_𝜎⁺_𝑏−1≤𝑏−1 and 𝑆_𝜎⁺_𝑏≥𝑏, the overshoot at time 𝜎⁺_𝑏 which is 𝑆_𝜎⁺_𝑏−𝑏 can take the values 0,1,2,…,𝑁−1, that is, 𝑆_𝜎⁺_𝑏∈{𝑏,𝑏+1,𝑏+2,…,𝑏+𝑁−1}. Similarly, when 𝜎⁻_𝑎<+∞, 𝑆_𝜎−𝑎∈{𝑎−𝑁+1,𝑎−𝑁+2,…,𝑎}, and when 𝜎_𝑎𝑏<+∞, 𝑆_𝜎𝑎𝑏∈{𝑎,𝑎−1,…,𝑎−𝑁+1}∪{𝑏,𝑏+1,…,𝑏+𝑁−1}. We put 𝑆⁺_𝑏=𝑆_𝜎⁺_𝑏, 𝑆⁻_𝑎=𝑆_𝜎−𝑎 and 𝑆_𝑎𝑏=𝑆_𝜎𝑎𝑏.

In the same way, we introduce the first overshooting times of the thresholds 𝑎<0 and 𝑏>0 (𝑎,𝑏 being now real numbers) for (𝑋_𝑡)_𝑡≥0: 𝜏⁻_𝑎=inf𝑡≥0∶𝑋_𝑡≤𝑎,𝜏⁺_𝑏=inf𝑡≥0∶𝑋_𝑡≥𝑏 (12)

as well as the first exit time from a bounded interval (𝑎,𝑏): 𝜏_𝑎𝑏=inf𝑡≥0∶𝑋_𝑡≤𝑎or𝑋_𝑡≥𝑏=inf𝑡≥0∶𝑋_𝑡∉(𝑎,𝑏) (13) with the similar convention that inf∅=+∞, and we set, when the corresponding time is finite, 𝑋⁻_𝑎=𝑋_𝜏−𝑎,𝑋⁺_𝑏=𝑋_𝜏⁺_𝑏,𝑋_𝑎𝑏=𝑋_𝜏𝑎𝑏. (14)

In this paper we provide a representation for the generating function of the joint distributions of the couples (𝜎⁻_𝑎,𝑆⁻_𝑎), (𝜎⁺_𝑏,𝑆⁺_𝑏), and (𝜎_𝑎𝑏,𝑆_𝑎𝑏). In particular, we derive simple expressions for the marginal distributions of 𝑆⁻_𝑎, 𝑆⁺_𝑏, and 𝑆_𝑎𝑏. We also obtain explicit expressions for the famous “ruin pseudoprobabilities” ℙ{𝜎⁻_𝑎<𝜎⁺_𝑏} and ℙ{𝜎⁺_𝑏<𝜎⁻_𝑎}. The main tool employed in this paper is the use of generating functions.

Taking that the limit as 𝜀 goes to zero, we retrieve the joint distributions of the couples (𝜏⁻_𝑎,𝑋⁻_𝑎) and (𝜏⁺_𝑏,𝑋⁺_𝑏) obtained in [10, 11]. Therein, we used Spitzer’s identity for deriving these distributions. Moreover, we obtain the joint distribution of the couple (𝜏_𝑎𝑏,𝑋_𝑎𝑏) which is a new and an important result for the study of pseudo-Brownian motion. In particular, we deduce the “ruin pseudo-probabilities” ℙ{𝜏⁻_𝑎<𝜏⁺_𝑏} and ℙ{𝜏⁺_𝑏<𝜏⁻_𝑎}; the results have been announced without any proof in a survey on pseudo-Brownian motion [13], after a conference held in Madrid (IWAP 2010).

In [11, 17, 18], the authors observed a curious fact concerning the pseudodistributions of 𝑋⁻_𝑎 and 𝑋⁺_𝑏: they are linear combinations of the Dirac distribution and its successive derivatives (in the sense of Schwarz distributions). For instance, ℙ𝑋⁺_𝑏∈d𝑧d𝑧=^𝑁−1_𝑗=0𝑏^𝑗𝛿𝑗!^𝑏(𝑗)(𝑧). (15)

The quantity 𝛿_𝑏^(𝑗) is to be understood as the functional acting on test functions 𝜙 according to ⟨𝛿_𝑏^(𝑗),𝜙⟩=(−1)^𝑗𝜙^(𝑗)(𝑏). The appearance of the 𝛿_𝑏^(𝑗)’s in (15), which is quite surprising for probabilists, can be better understood thanks to the discrete approach. Indeed, the 𝛿_𝑏^(𝑗)’s come from the location at the overshooting time of 𝑏 for the normalized pseudorandom walk: the location takes place in the “cluster” of points 𝑏,𝑏+𝜀,𝑏+2𝜀,…,𝑏+(𝑁−1)𝜀.

In order to facilitate the reading of the paper, we have divided it into three parts: Part I—some properties of the pseudorandom walk Part II—first overshooting time of a single threshold Part III—first exit time from a bounded interval.

The reader will find a list of notations in Table 2 which is postponed to the end of the paper.

2. Part I—Some Properties of the Pseudorandom Walk

2.1. Pseudodistribution of 𝜉₁ and 𝑆_𝑛

We consider the pseudorandom walk (𝑆_𝑛)_𝑛∈ℕ related to a family of real parameters {𝑝_𝑘,𝑘∈{−𝑁,…,𝑁}} satisfying 𝑝_𝑘=𝑝_−𝑘 for any 𝑘∈{1,…,𝑁} and ∑^𝑁_𝑘=−𝑁𝑝_𝑘=1. Let us recall that the infinitesimal generator associated with (𝑆_𝑛)_𝑘≥0 is defined by 𝒢_𝑆𝑓𝑗=^𝑁_𝑘=−𝑁𝑝_𝑘𝑓𝑗=𝑝𝑗+𝑘−𝑓₀−1𝑓𝑗+^𝑁_𝑘=1𝑝_𝑘𝑓𝑗+𝑘+𝑓𝑗−𝑘. (16)

In this section, we look for the values of 𝑝_𝑘, 𝑘∈{−𝑁,…,𝑁}, for which the infinitesimal generator 𝒢 is of the form (5). Next, we provide several properties for the corresponding pseudorandom walk.

Suppose that 𝑓 can be extended into an analytical function 𝑓. In this case, we can expand 𝑓𝑗+𝑘+𝑓𝑗−𝑘=2^∞_ℓ=0𝑘^2ℓ()!𝑓2ℓ^(2ℓ)𝑗. (17) Therefore, 𝒢𝑓𝑗=𝑝₀−1𝑓𝑗+2^𝑁_𝑘=1𝑝_𝑘^∞_ℓ=0𝑘^2ℓ()!𝑓2ℓ^(2ℓ)𝑗=𝑝₀+2^𝑁_𝑘=1𝑝_𝑘−1𝑓𝑗+2^∞_ℓ=1^𝑁_𝑘=1𝑘^2ℓ𝑝_𝑘𝑓^(2ℓ)𝑗()!.2ℓ (18) Since 𝑝₀+2∑^𝑁_𝑘=1𝑝_𝑘=1, we see that the expression (5) of 𝒢 holds if and only if the 𝑝_𝑘’s satisfy the equations ^𝑁_𝑘=1𝑘^2ℓ𝑝_𝑘=0for1≤ℓ≤𝑁−1,_𝑁𝑘=1𝑘^2𝑁𝑝_𝑘=12𝜅_𝑁𝑐(2𝑁)!. (19)

Proposition 1. The numbers 𝑝_𝑘, 𝑘∈{1,…,𝑁}, satisfying (19), are given by 𝑝_𝑘=(−1)^𝑘−1𝑐.2𝑁𝑘+𝑁 (20) In particular, 𝑝_𝑁=𝜅_𝑁𝑐.

Proof. First, we recall that the solution of a Vandermonde system of the form ∑^𝑁_𝑘=1𝑎^ℓ_𝑘𝑥_𝑘=𝛼_ℓ, 1≤ℓ≤𝑁, is given by 𝑥_𝑘=𝐴_𝑘^{𝑎1,…,𝑎𝑁}_{𝛼1,…,𝛼𝑁}𝐴𝑎₁,…,𝑎_𝑁,1≤𝑘≤𝑁 (21) with 𝐴𝑎₁,…,𝑎_𝑁=𝑎₁⋯𝑎_𝑁𝑎²₁⋯𝑎²_𝑁𝑎⋮⋮_𝑁1⋯𝑎^𝑁_𝑁 (22) and, for any 𝑘∈{1,…,𝑁}, 𝐴_𝑘𝑎₁,…,𝑎_𝑁𝛼₁,…,𝛼_𝑁=𝑎₁⋯𝑎_𝑘−1𝛼₁𝑎_𝑘+1⋯𝑎_𝑁𝑎²₁⋯𝑎²_𝑘−1𝛼₂𝑎²_𝑘+1⋯𝑎²_𝑁𝑎⋮⋮⋮⋮⋮_𝑁1⋯𝑎^𝑁_𝑘−1𝛼_𝑁𝑎^𝑁_𝑘+1⋯𝑎^𝑁_𝑁. (23) In the notation of 𝐴_𝑘 and that of forthcoming determinants, we adopt the convention that when the index of certain entries in the determinant lies out of the range of 𝑘, the corresponding column is discarded. That is, for 𝑘=1 and 𝑘=𝑁, the respective determinants write 𝐴₁𝑎₁,…,𝑎_𝑁𝛼₁,…,𝛼_𝑁=𝛼₁𝑎₂⋯𝑎_𝑁𝛼⋮⋮⋮_𝑁𝑎^𝑁₂⋯𝑎^𝑁_𝑁,𝐴_𝑁𝑎₁,…,𝑎_𝑁𝛼₁,…,𝛼_𝑁=𝑎₁⋯𝑎_𝑁−1𝛼₁𝑎⋮⋮⋮_𝑁1⋯𝑎^𝑁_𝑁−1𝛼_𝑁. (24) It is well-known that, for any 𝑘∈{1,…,𝑁}, 𝐴𝑎₁,…,𝑎_𝑁=_{1≤𝑗≤𝑁}𝑎_{𝑗1≤ℓ<𝑚≤𝑁}𝑎_𝑚−𝑎_ℓ=(−1)^𝑘+𝑁_{1≤𝑗≤𝑁}𝑎_{𝑗1≤𝑗≤𝑁𝑗≠𝑘}𝑎_𝑘−𝑎_𝑗×_{1≤ℓ<𝑚≤𝑁ℓ,𝑚≠𝑘}𝑎_𝑚−𝑎_ℓ. (25) In the particular case where 𝛼_ℓ=0 for 1≤ℓ≤𝑁−1, we have, for any 𝑘∈{1,…,𝑁}, that 𝐴_𝑘𝑎₁,…,𝑎_𝑁𝛼₁,…,𝛼_𝑁=(−1)^𝑘+𝑁𝛼_𝑁×𝑎₁⋯𝑎_𝑘−1𝑎_𝑘+1⋯𝑎_𝑁𝑎²₁⋯𝑎²_𝑘−1𝑎²_𝑘+1⋯𝑎²_𝑁𝑎⋮⋮⋮⋮_1𝑁−1⋯𝑎^𝑁−1_𝑘−1𝑎^𝑁−1_𝑘+1⋯𝑎_𝑁^𝑁−1.=(−1)^𝑘+𝑁𝛼_{𝑁1≤𝑗≤𝑁𝑗≠𝑘}𝑎_{𝑗1≤ℓ<𝑚≤𝑁ℓ,𝑚≠𝑘}𝑎_𝑚−𝑎_ℓ. (26) Therefore, the solution simply writes 𝑥_𝑘=𝛼_𝑁𝑎_𝑘∏_{1≤𝑗≤𝑁𝑗≠𝑘}𝑎_𝑘−𝑎_𝑗,1≤𝑘≤𝑁. (27)
Now, we see that system (19) is a Vandermonde system with the choices 𝑎_𝑘=𝑘², 𝑥_𝑘=𝑝_𝑘, and 𝛼_ℓ=0 for 1≤ℓ≤𝑁−1, 𝛼_𝑁=𝜅_𝑁𝑐(2𝑁)!/2. With these settings at hands, we explicitly have 𝑎_{𝑘1≤𝑗≤𝑁𝑗≠𝑘}𝑎_𝑘−𝑎_𝑗=𝑘²_{1≤𝑗≤𝑁𝑗≠𝑘}𝑘²−𝑗²=𝑘²_{1≤𝑗≤𝑁𝑗≠𝑘}𝑘−𝑗_{1≤𝑗≤𝑁𝑗≠𝑘}=1𝑘+𝑗2(−1)^𝑁−𝑘(𝑁+𝑘)!(𝑁−𝑘)! (28)
and the result of Proposition 1 ensues.

Finally, the value of 𝑝₀ is obtained as follows: by using the fact that ∑^2𝑁_𝑘=0(−1)^𝑘_𝑘^2𝑁=0, 𝑝₀=1−_{−𝑁≤𝑘≤𝑁𝑘≠0}𝑝_𝑘=1+𝑐_{−𝑁≤𝑘≤𝑁𝑘≠0}(−1)^𝑘𝑁+()2𝑁𝑘+𝑁=1−𝑐2𝑁−1^𝑁𝑐^2𝑁_𝑘=0(−1)^𝑘𝑘𝑁.2𝑁=1−𝑐2𝑁 (29) We find it interesting to compute the cumulative sums of the 𝑝_𝑗’s: for 𝑘∈{−𝑁,…,𝑁}, ^𝑘_𝑗=−𝑁𝑝_𝑗=^𝑘_𝑗=−𝑁𝛿_𝑗0+(−1)^𝑗−1𝑐2𝑁𝑗+𝑁=𝟙_{𝑘≥0}+(−1)^𝑁−1𝑐^𝑘+𝑁_𝑗=0(−1)^𝑗𝑗.2𝑁 (30) The last displayed sum is classical and easy to compute by appealing to Pascal’s formula which leads to a telescopic sum: ^𝑘+𝑁_𝑗=0(−1)^𝑗𝑗=2𝑁_{𝑘+𝑁𝑗=0}(−1)^𝑗−()2𝑁−1𝑗−1−1^𝑗+1𝑗=()2𝑁−1−1^𝑘+𝑁.2𝑁−1𝑘+𝑁 (31) Thus, for 𝑘∈{−𝑁,…,𝑁}, ^𝑘_𝑗=−𝑁𝑝_𝑗=𝟙_{𝑘≥0}+(−1)^𝑘−1𝑐.2𝑁−1𝑘+𝑁 (32) Observe that this sum is nothing but ℙ{𝜉₁≤𝑘}. Next, we compute the total sum of the |𝑝_𝑗|’s: by using the fact that ∑^2𝑁_𝑘=0𝑘^2𝑁=4^𝑁, ^𝑁_𝑘=−𝑁𝑝_𝑘=𝑁1−𝑐2𝑁+_{−𝑁≤𝑘≤𝑁𝑘≠0}𝑐=𝑁42𝑁𝑘+𝑁1−𝑐2𝑁+𝑐^𝑁−𝑁2𝑁=𝑐4^𝑁𝑁−1+21−𝑐2𝑁⁺. (33) As previously mentioned, there is an interpretation to this sum: this is the total variation of the pseudodistribution of 𝜉₁. We can also explicitly determine the generating function of 𝜉₁: for any 𝜁∈ℂ^∗, 𝔼𝜁^𝜉1=^𝑁_𝑘=−𝑁𝑝_𝑘𝜁^𝑘=1+𝑐^𝑁_𝑘=−𝑁(−1)^𝑘−1𝜁2𝑁𝑘+𝑁^𝑘=1+(−1)^𝑁−1𝑐𝜁^𝑁2𝑁_𝑘=0(−1)^𝑘𝑘𝜁2𝑁_𝑘=1+𝜅_𝑁𝑐(1−𝜁)^2𝑁𝜁^𝑁. (34) We sum up below the results we have obtained concerning the pseudodistribution of 𝜉₁.

Proposition 2. The pseudodistribution of 𝜉₁ is determined, for 𝑘∈{−𝑁,…,𝑁}, by ℙ𝜉₁=𝑘=𝛿_𝑘0+(−1)^𝑘−1𝑐,2𝑁𝑘+𝑁 (35) or, equivalently, by ℙ𝜉₁≤𝑘=𝟙_{𝑘≥0}+(−1)^𝑘−1𝑐.2𝑁−1𝑘+𝑁 (36) The total variation of the pseudodistribution of 𝜉₁ is given by ℙ_𝜉1=1+𝑐4^𝑁𝑁1−22𝑁𝑖𝑓0<𝑐≤_𝑁2𝑁,𝑐4^𝑁1−1𝑖𝑓𝑐≥_𝑁2𝑁. (37) The generating function of 𝜉₁ is given, for any 𝜁∈ℂ^∗, by 𝔼𝜁^𝜉1=1+𝜅_𝑁𝑐(1−𝜁)^2𝑁𝜁^𝑁. (38) In particular, the Fourier transform of 𝜉₁ admits the following expression: for any 𝜃∈[0,2𝜋], by 𝔼e^{𝑖𝜃𝜉1}=1−𝑐4^𝑁sin^2𝑁𝜃2. (39)

Remark 3. For 𝑐=1/_𝑁^2𝑁, we have ℙ{𝜉₁=0}=0; that is, the pseudorandom walk does not stay at its current location. If 0<𝑐≤1/^2𝑁+1_𝑁+1, it can be easily seen, by using the identity _𝑁^2𝑁+^2𝑁_𝑁+1=^2𝑁+1_𝑁+1, that ℙ{𝜉₁=0}≥ℙ{𝜉₁=1}>0. On the other hand, for any 𝑐>0, it is clear that ℙ{𝜉₁=1}>|ℙ{𝜉₁=2}|>⋯>|ℙ{𝜉₁=𝑁}|. In Table 1 and Figures 1 and 2, we provide some numerical values and (rescaled) profiles of the pseudodistribution of 𝜉₁ for 𝑁=3 and 𝑁=4 and several values of 𝑐.

Table 1 

Some numerical values in the cases 𝑁∈{3,4}.

Figure 1 

𝑁=3, 𝑐∈{1,1/32,1/64}.

Figure 2 

𝑁=4, 𝑐∈{1,1/128,1/256}.

In the sequel, we will use the total variation of 𝜉₁ as an upper bound which we call 𝑀₁: 𝑀₁=1+𝑐4^𝑁𝑁−22𝑁if10<𝑐≤_𝑁2𝑁,𝑐4^𝑁−1if1𝑐≥_𝑁2𝑁. (40) Set 𝔣(𝜃)=𝔼(e^{𝑖𝜃𝜉1}) for any 𝜃∈[0,2𝜋]. We notice that 𝔣(𝜃)∈[1−𝑐4^𝑁,1] and, more precisely, ‖𝔣‖_∞=sup_{[]𝜃∈0,2𝜋}|𝔣(𝜃)|=max1−𝑐4^𝑁=1,1if10<𝑐≤2^2𝑁−1,𝑐4^𝑁−1if1𝑐≥2^2𝑁−1. (41) Let us denote this bound by 𝑀_∞: 𝑀_∞=1if10<𝑐≤2^2𝑁−1,𝑐4^𝑁−1if1𝑐≥2^2𝑁−1. (42) In view of (40) and (42), since _𝑁^2𝑁≤2^2𝑁−1, we see that 𝑀₁≥𝑀_∞≥1.

Proposition 4. The pseudodistribution of 𝑆_𝑛 is given, for any 𝑘∈{−𝑁𝑛,…,𝑁𝑛}, by ℙ𝑆_𝑛=𝑘=(−1)^𝑘𝑛_ℓ=0(−𝑐)^ℓ𝑛ℓ.2𝑁ℓ𝑘+𝑁ℓ (43) Actually, the foregoing sum is taken over the ℓ such that ℓ≥|𝑘|/𝑁. We also have that ℙ𝑆_𝑛≤𝑘=𝟙_{𝑘≥0}+(−1)^𝑘𝑛_ℓ=1(−𝑐)^ℓ𝑛ℓ.2𝑁ℓ−1𝑘+𝑁ℓ (44)

Proof. By the independence of the 𝜉_𝑗’s which have the same pseudoprobability distribution, we plainly have that 𝔼e^{𝑖𝜃𝑆𝑛}=𝔣(𝜃)^𝑛=1−𝑐4^𝑁sin^2𝑁𝜃2^𝑛. (45) Hence, by inverse Fourier transform, we extract that ℙ𝑆_𝑛=1=𝑘2𝜋₀^2𝜋𝔣(𝜃)^𝑛e^{−𝑖𝑘𝜃}d𝜃=^𝑛_ℓ=0−4^𝑁𝑐^ℓ𝑛ℓ12𝜋₀^2𝜋sin^2𝑁ℓ𝜃2e^{−𝑖𝑘𝜃}d𝜃. (46)(47) By writing sin(𝜃/2)=(e^𝑖𝜃/2−e^{−𝑖𝜃/2})/(2𝑖), we get for the integral lying in (47) that 12𝜋₀^2𝜋sin^2𝑁ℓ𝜃2e^{−𝑖𝑘𝜃}d𝜃=12𝜋₀^2𝜋e^𝑖𝜃/2−e^{−𝑖𝜃/2}2𝑖^2𝑁ℓe^{−𝑖𝑘𝜃}d𝜃=(−1)^𝑁ℓ(2𝜋)4^𝑁ℓ₀^{2𝜋2𝑁ℓ}_𝑚=0(−1)^𝑚𝑚2𝑁ℓe^{𝑖(𝑚−𝑘−𝑁ℓ)𝜃}d𝜃=(−1)^𝑁ℓ4^{𝑁ℓ2𝑁ℓ}_𝑚=0(−1)^𝑚𝑚2𝑁ℓ12𝜋₀^2𝜋e^{𝑖(𝑚−𝑘−𝑁ℓ)𝜃}d𝜃=(−1)^𝑘4^𝑁ℓ.2𝑁ℓ𝑘+𝑁ℓ (48) By plugging (48) into (47), we derive (43). Next, we write, for 𝑘∈{−𝑁𝑛,…,𝑁𝑛}, that ℙ𝑆_𝑛≤𝑘=^𝑘_{𝑗=−𝑁𝑛}ℙ𝑆_𝑛==𝑗_𝑛ℓ=0(−𝑐)^ℓ𝑛ℓ^{𝑘∧(𝑁ℓ)()∨()𝑗=−𝑁𝑛−𝑁ℓ}(−1)^𝑗=2𝑁ℓ𝑗+𝑁ℓ_𝑛ℓ=0(−𝑐)^ℓ𝑛ℓ^{𝑘∧(𝑁ℓ)𝑗=−𝑁ℓ}(−1)^𝑗.2𝑁ℓ𝑗+𝑁ℓ (49) If 𝑘<0, then the term in sum (49) corresponding to ℓ=0 vanishes and ℙ𝑆_𝑛≤𝑘=^𝑛_ℓ=1(−𝑐)^ℓ𝑛ℓ^{𝑘𝑗=−𝑁ℓ}(−1)^𝑗.2𝑁ℓ𝑗+𝑁ℓ (50) The second sum in the foregoing equality is easy to compute: ^𝑘_{𝑗=−𝑁ℓ}(−1)^𝑗=2𝑁ℓ𝑗+𝑁ℓ_{𝑘𝑗=−𝑁ℓ}(−1)^𝑗−()2𝑁ℓ−1𝑗+𝑁ℓ−1−1^𝑗+1=()2𝑁ℓ−1𝑗+𝑁ℓ−1^𝑘.2𝑁ℓ−1𝑘+𝑁ℓ (51) If 𝑘≥0, then the term in sum (49) corresponding to ℓ=0 is 1 and ℙ𝑆_𝑛≤𝑘=1+^𝑛_ℓ=1(−𝑐)^ℓ𝑛ℓ^{𝑘∧(𝑁ℓ)𝑗=−𝑁ℓ}(−1)^𝑗.2𝑁ℓ𝑗+𝑁ℓ (52) By using the convention that ^𝛼_𝛽=0 if 𝛽>𝛼, we see that the second sum above also coincides with (51). Formula (44) ensues in both cases.

Proposition 5. The upper bound below holds true: for any positive integer 𝑛 and any integer 𝑘, ℙ𝑆_𝑛=𝑘≤ℙ𝑆_2𝑛=0≤𝑀^𝑛_∞. (53) Assume that 0<𝑐≤1/2^2𝑁−1. The asymptotics below holds true: for any 𝛿∈(0,1/(2𝑁)), ℙ𝑆_𝑛=0=_𝑛→+∞𝒪1𝑛^𝛿. (54)

Proof. Let us introduce the usual norms of any suitable function 𝜙: 𝜙₁=12𝜋₀^2𝜋𝜙(𝜃)d𝜙𝜃,₂=12𝜋₀^2𝜋𝜙(𝜃)²d𝜙𝜃,_∞=sup_{[]𝜃∈0,2𝜋}𝜙(𝜃) (55)
and recall the elementary inequalities ‖𝜙‖₁≤‖𝜙‖₂≤‖𝜙‖_∞.
It is clear from (46) that, for any integer 𝑘, ℙ𝑆_𝑛≤1=𝑘2𝜋₀^2𝜋|𝔣(𝜃)|^𝑛d𝜃=𝔣^𝑛₁≤𝔣^𝑛₂=12𝜋₀^2𝜋𝔣(𝜃)^2𝑛d𝜃=ℙ𝑆_2𝑛≤‖𝔣‖=0_𝑛∞=𝑀^𝑛_∞. (56) This proves (53). Next, by (46), since 𝔣(2𝜋−𝜃)=𝔣(𝜃), we have, for any 𝜀∈(0,𝜋), that ℙ𝑆_𝑛=1=02𝜋₀^2𝜋𝔣(𝜃)^𝑛d1𝜃=𝜋^𝜋₀𝔣(𝜃)^𝑛d𝜃=1𝜋^𝜀₀𝔣(𝜃)^𝑛d𝜃+^𝜋_𝜀𝔣(𝜃)^𝑛d𝜃. (57) The assumption 0<𝑐<1/2^2𝑁−1 entails that |𝔣(𝜃)|<1 for any 𝜃∈(0,𝜋). We see that |𝔣(𝜃)|≤1 on [0,𝜀], and |𝔣(𝜃)|≤|𝔣(𝜀)| on [𝜀,𝜋] for any 𝜀∈(0,𝜋). Hence, ℙ𝑆_𝑛=0≤𝜀+|𝔣(𝜀)|^𝑛. (58) Now, choose 𝜀=1/𝑛^𝛿 for a positive 𝛿. We have that ln|𝔣(𝜀)|^𝑛=𝑛ln1−𝑐4^𝑁sin^2𝑁12𝑛^𝛿∼−𝑐𝑛^{1−2𝑁𝛿} (59)
which clearly entails, for large enough 𝑛, |𝔣(𝜀)|^𝑛≤exp(−(𝑐/2)𝑛^{1−2𝑁𝛿}).Thus, if 𝛿∈(0,1/(2𝑁)), |𝔣(𝜀)|^𝑛=𝑜(𝜀) which proves (54).
If 𝑐=1/2^2𝑁−1, 𝔣(𝜃)=1−2sin^2𝑁(𝜃/2). In this case, the same holds true upon splitting the integral ∫^𝜋₀ into ∫^𝜀₀+∫_𝜀^𝜋−𝜀+∫^𝜋_𝜋−𝜀.

Remark 6. A better estimate for |ℙ{𝑆_𝑛=𝑘}| can be obtained in the same way: ∀𝑘∈ℤ,ℙ𝑆_𝑛≤ℙ𝑆=𝑘^𝑛=0if𝑛iseven,𝑀_∞ℙ𝑆_𝑛−1=0if𝑛isodd. (60) Nevertheless, we will not use it. We also have the following inequality for the total variation of 𝑆_𝑛: ℙ_𝑆𝑛=^𝑁𝑛_{𝑘=−𝑁𝑛}ℙ𝑆_𝑛=𝑘≤ℙ_𝜉1^𝑛=𝑀^𝑛₁. (61)

Proposition 7. For any bounded function 𝐹 defined on ℤ^𝑛, 𝔼𝐹𝑆₁,…,𝑆_𝑛≤‖𝐹‖_∞𝑀^𝑛₁. (62)

Proof. Recall that we set 𝑝_𝑘=ℙ{𝜉₁=𝑘} for any 𝑘∈{−𝑁,…,𝑁}. We extend these settings by putting 𝑝_𝑘=0 for 𝑘∈ℤ⧵{−𝑁,…,𝑁}. We have that 𝔼𝐹𝑆₁,…,𝑆_𝑛=_{𝑘1,…,𝑘𝑛∈ℤ𝑛}𝐹𝑘₁,…,𝑘_𝑛ℙ𝑆₁=𝑘₁,…,𝑆_𝑛=𝑘_𝑛≤‖𝐹‖_{∞𝑘1,…,𝑘𝑛∈ℤ𝑛}𝑝_𝑘1𝑝_{𝑘2−𝑘1}⋯𝑝_{𝑘𝑛−𝑘𝑛−1}. (63) The foregoing sum can be easily evaluated as follows: _{𝑘1,…,𝑘𝑛∈ℤ𝑛}𝑝_𝑘1𝑝_{𝑘2−𝑘1}⋯𝑝_{𝑘𝑛−𝑘𝑛−1}=_𝑘1∈ℤ𝑝_{𝑘1𝑘2∈ℤ}𝑝_{𝑘2−𝑘1}⋯_{𝑘𝑛∈ℤ}𝑝_{𝑘𝑛−𝑘𝑛−1}⋯=_𝑘∈ℤ𝑝_𝑘^𝑛=𝑀^𝑛₁ (64) which proves (62).

2.2. Generating Function of 𝑆_𝑛

Let us introduce the generating functions, defined for complex numbers 𝑧, 𝜁, by 𝐺_𝑘(𝑧)=_𝑛∈ℕℙ𝑆_𝑛=𝑘𝑧^𝑛=_{|𝑘|𝑛∈ℕ∶𝑛≥/𝑁}ℙ𝑆_𝑛=𝑘𝑧^𝑛for𝐺()𝑘∈ℤ,𝑧,𝜁=_{𝑘∈ℤ,𝑛∈ℕ}ℙ𝑆_𝑛=𝑘𝜁^𝑘𝑧^𝑛=_{|𝑘|𝑘∈ℤ,𝑛∈ℕ∶≤𝑁𝑛}ℙ𝑆_𝑛=𝑘𝜁^𝑘𝑧^𝑛. (65) We first study the problem of convergence of the foregoing series. We start from _{𝑘∈ℤ,𝑛∈ℕ}ℙ𝑆_𝑛=𝑘𝜁^𝑘𝑧^𝑛≤_{|𝑘|𝑘∈ℤ,𝑛∈ℕ∶≤𝑁𝑛}𝜁^𝑘𝑀_∞𝑧^𝑛. (66) If 𝜁≠0 and |𝜁|≠1, then _{|𝑘|𝑘∈ℤ,𝑛∈ℕ∶≤𝑁𝑛}𝜁^𝑘𝑀_∞𝑧^𝑛=^∞_𝑛=01−𝜁^𝑁𝑛+1𝜁+𝜁1−1−1/^𝑁𝑛+1𝜁𝑀1−1/−1_∞𝑧^𝑛=11−𝜁^∞_𝑛=0𝑀_∞𝑧^𝑛−𝜁^∞_𝑛=0𝑀_∞𝑧𝜁^𝑁𝑛+11−1/𝜁^∞_𝑛=0𝑀_∞𝑧^𝑛−1𝜁^∞_𝑛=0𝑀_∞𝑧𝜁^𝑁𝑛−^∞_𝑛=0𝑀_∞𝑧^𝑛. (67) If we choose 𝑧, 𝜁 such that |𝑀_∞𝑧|<1, |𝑀_∞𝑧𝜁^𝑁|<1 and |𝑀_∞𝑧/𝜁^𝑁|<1 (which is equivalent to |𝑧|<1/𝑀_∞×[min(|𝜁|,1/|𝜁|)]^𝑁, or ^𝑁|𝑀_∞𝑧|<|𝜁|<1/^𝑁|𝑀_∞𝑧|), then the double sum defining the function 𝐺(𝑧,𝜁) is absolutely summable. If |𝜁|=1, then _{|𝑘|𝑘∈ℤ,𝑛∈ℕ∶≤𝑁𝑛}𝜁^𝑘𝑀_∞𝑧^𝑛=^∞_𝑛=0(2𝑁𝑛+1)𝑀_∞𝑧^𝑛. (68)

If we choose 𝑧 such that |𝑀_∞𝑧|<1, then the same conclusion holds.

Now, we have that 𝐺(𝑧,𝜁)=_{𝑛∈ℕ𝑘∈ℤ}ℙ𝑆_𝑛=𝑘𝜁^𝑘𝑧^𝑛=_𝑛∈ℕ𝔼𝜁^𝑆𝑛𝑧^𝑛=_𝑛∈ℕ𝑧𝔼𝜁^𝜉1𝑛=11−𝑧𝔼𝜁^𝜉1 (69) and, thanks to (38), we can state the following result.

Proposition 8. The double generating function of the ℙ{𝑆_𝑛=𝑘}, 𝑘∈ℤ, 𝑛∈ℕ, is given, for any complex numbers 𝑧, 𝜁 such that ^𝑁|𝑀_∞𝑧|<|𝜁|<1/^𝑁|𝑀_∞𝑧|, by 𝐺()=𝜁𝑧,𝜁^𝑁(1−𝑧)𝜁^𝑁−𝜅_𝑁𝑐𝑧(1−𝜁)^2𝑁. (70) In particular, for any 𝜃∈[0,2𝜋] and 𝑧∈ℂ such that |𝑧|<1/𝑀_∞, 𝐺𝑧,e^𝑖𝜃=11−𝑧+𝑐4^𝑁𝑧sin^2𝑁().𝜃/2 (71)

On the other hand, 𝐺(𝑧,𝜁)=_{𝑘∈ℤ𝑛∈ℕ}ℙ𝑆_𝑛=𝑘𝑧^𝑛𝜁^𝑘=_𝑘∈ℤ𝐺_𝑘(𝑧)𝜁^𝑘. (72) By substituting 𝜁=e^𝑖𝜃 in the foregoing equality, we get the Fourier series of the function 𝜃↦𝐺(𝑧,e^𝑖𝜃): 𝐺𝑧,e^𝑖𝜃=_𝑘∈ℤ𝐺_𝑘(𝑧)e^𝑖𝑘𝜃 (73) from which we extract the sequence of the coefficients (𝐺_𝑘(𝑧))_𝑘∈ℕ. Indeed, since ℙ{𝑆_𝑛=−𝑘}=ℙ{𝑆_𝑛=𝑘}, we have that 𝐺_𝑘(𝑧)=𝐺_−𝑘(𝑧) and 𝐺_𝑘(𝑧)=12𝜋₀^2𝜋𝐺𝑧,e^𝑖𝜃e^{−𝑖𝑘𝜃}d𝜃=12𝜋₀^2𝜋𝐺𝑧,e^𝑖𝜃e^𝑖𝑘𝜃d𝜃=12𝜋𝑖_𝒞𝐺(𝑧,𝜁)𝜁^𝑘−1d𝜁, (74) where 𝒞 is the circle of radius 1 centered at the origin and counter clockwise orientated. Then, referring to (70), we obtain, for any 𝑧∈ℂ satisfying |𝑧|<1/𝑀_∞, that 𝐺_𝑘(𝑧)=12𝜋𝑖_𝒞𝜁^{𝑘+𝑁−1}𝑃(𝑧,𝜁)d𝜁, (75) where 𝑃(𝑧,𝜁) is the polynomial given by 𝑃(𝑧,𝜁)=(1−𝑧)𝜁^𝑁−𝜅_𝑁𝑐𝑧(𝜁−1)^2𝑁. (76)

We are looking for the roots of 𝜁↦𝑃(𝑧,𝜁) which lie inside the circle 𝒞. For this, we introduce the 𝑁th roots of 𝜅_𝑁: 𝜃_𝑗=−e^{𝑖((2𝑗−1)/𝑁)𝜋} for 1≤𝑗≤𝑁; 𝜃^𝑁_𝑗=𝜅_𝑁.

From now on, in order to simplify the expression of the roots of 𝑃, we make the assumption that 𝑧 is a real number lying in (0,1/𝑀_∞) (and then 𝑧∈(0,1)). The roots of 𝜁↦𝑃(𝑧,𝜁) are those of the equations 𝜁²−2[1+𝜃_𝑗𝑤(𝑧)]𝜁+1=0, 1≤𝑗≤𝑁, where 𝑤(𝑧)=^𝑁√1−𝑧2^𝑁√.𝑐𝑧 (77) They can be written as 𝑢_𝑗(𝑧)=1+𝜃_𝑗𝑤(𝑧)−𝜃_𝑗𝑤(𝑧)𝑎_𝑗(𝑧)+𝑖𝜖_𝑗𝑏_𝑗(𝑧𝑣),1≤𝑗≤𝑁,_𝑗(𝑧)=1+𝜃_𝑗𝑤(𝑧)+𝜃_𝑗𝑤(𝑧)𝑎_𝑗(𝑧)+𝑖𝜖_𝑗𝑏_𝑗(𝑧),1≤𝑗≤𝑁, (78) with 𝜖_𝑗=sgnsin2𝑗−1𝑁𝜋withtheconventionthat(0),𝑎sgn=0_𝑗(𝑧)=1√2𝑤(𝑧)²−4cos2𝑗−1𝑁𝜋𝑤(𝑧)(𝑧)+4+𝑤−2cos2𝑗−1𝑁𝜋^1/2,𝑏_𝑗(𝑧)=1√2𝑤(𝑧)²−4cos2𝑗−1𝑁𝜋𝑤(𝑧)(𝑧)+4−𝑤+2cos2𝑗−1𝑁𝜋^1/2. (79)

We notice that 𝑎_𝑗(𝑧)𝑏_𝑗(𝑧)=|sin(((2𝑗−1)/𝑁)𝜋)|. Because of the last coefficient 1 in the polynomial 𝜁²−2[1+𝜃_𝑗𝑤(𝑧)]𝜁+1, it is clear that the roots 𝑢_𝑗(𝑧) and 𝑣_𝑗(𝑧) are inverse: 𝑣_𝑗(𝑧)=1/𝑢_𝑗(𝑧).

Let us check that |𝑢_𝑗(𝑧)|<1<|𝑣_𝑗(𝑧)| for any 𝑗∈{1,…,𝑁}. Straightforward computations yield that 𝑢_𝑗(𝑧)²=𝐴_𝑗(𝑧)−𝐵_𝑗(𝑧),𝑣_𝑗(𝑧)²=𝐴_𝑗(𝑧)+𝐵_𝑗(𝑧), (80) where 𝐴_𝑗(𝑧)=𝑤(𝑧)²(𝑧)×+𝑤𝑤(𝑧)²−4cos2𝑗−1𝑁𝜋𝑤(𝑧)+4−2cos2𝑗−1𝑁𝜋𝐵+1,_𝑗(𝑧)=2𝑤(𝑧)×𝑤(𝑧)−cos2𝑗−1𝑁𝜋𝑎_𝑗(𝑧)+sin2𝑗−1𝑁𝜋𝑏_𝑗(𝑧). (81) Since 𝑣_𝑗(𝑧)=1/𝑢_𝑗(𝑧), checking that |𝑢_𝑗(𝑧)|<1<|𝑣_𝑗(𝑧)| is equivalent to checking that |𝑢_𝑗(𝑧)|²<|𝑣_𝑗(𝑧)|²; that is, 𝐵_𝑗(𝑧)>0. If sin(((2𝑗−1)/𝑁)𝜋)≠0, we have 𝑎_𝑗(𝑧)𝑏_𝑗(𝑧)≠0, 𝑎_𝑗(𝑧)=|sin(((2𝑗−1)/𝑁)𝜋)|/𝑏_𝑗(𝑧) and then 𝐵_𝑗(𝑧)=2𝑎_𝑗(𝑧)𝑤(𝑧)𝑤(𝑧)−cos2𝑗−1𝑁𝜋+𝑏_𝑗(𝑧)²=𝑎_𝑗(𝑧)𝑤(𝑧)×𝑤(𝑧)²−4cos2𝑗−1𝑁𝜋𝑤(𝑧)+4+𝑤(𝑧)>0. (82)

If sin(((2𝑗−1)/𝑁)𝜋)=0 (which happens only when 𝑁 is odd and 𝑗=(𝑁+1)/2), 𝑎_𝑗(𝑧)=𝑤(𝑧)+2, 𝑏_𝑗(𝑧)=0 and then 𝐵_𝑗=2𝑎_𝑗(𝑧)𝑤(𝑧)(𝑤(𝑧)+1)>0.

The above discussion ensures that the roots we are looking for (i.e., those lying inside 𝒞) are 𝑢_𝑗(𝑧), 1≤𝑗≤𝑁; we discard the 𝑣_𝑗(𝑧)’s.

Remark 9. We notice that lim_𝑧→1−𝑎_𝑗(𝑧)=√2sin2𝑗−1𝜋,2𝑁lim_𝑧→1−𝑏_𝑗(𝑧)=𝜖_𝑗√2cos2𝑗−1𝜋2𝑁 (83) and then lim_𝑧→1−𝜃_𝑗𝑎_𝑗(𝑧)+𝑖𝜖_𝑗𝑏_𝑗(𝑧)=√2𝜑_𝑗, (84) where we set 𝜑_𝑗=−𝑖e^{𝑖((2𝑗−1)/2𝑁)𝜋}. The 𝜑_𝑗, 1≤𝑗≤𝑁, are the (2𝑁)th roots of 𝜅_𝑁 with positive real part: 𝜑_𝑗^2𝑁=𝜅_𝑁 and ℜ(𝜑_𝑗)>0. As a result, we derive the asymptotics, which will be used further, 𝑢_𝑗(𝑧)=1+𝜀_𝑗(𝑧)with𝜀_𝑗(𝑧)∼_𝑧→1−−_𝑗𝜑_2𝑁√𝑐^2𝑁√1−𝑧=𝒪^2𝑁√1−𝑧. (85)

Example 10. For 𝑁=1, the roots explicitly write as 𝑢₁(𝑧)=1+𝑤(𝑧)−𝑎₁(𝑧)𝑤(𝑧),𝑣₁(𝑧)=1𝑢₁(𝑧) (86) with 𝑤(𝑧)=1−𝑧2𝑐𝑧,𝑎₁(𝑧)=𝑤(𝑧)+2. (87) If 𝑐=1/2, it can be simplified into 𝑢₁(𝑧)=1−√1−𝑧²𝑧. (88) For 𝑁=2, the roots explicitly write as 𝑢₁(𝑧)=1−𝑏₁(𝑧)𝑤(𝑧)−𝑖𝑤(𝑧)−𝑎₁(𝑧)𝑤(𝑧,𝑢₂)(𝑧)=𝑢₁(𝑧),𝑣₁(𝑧)=1𝑢₁(𝑧),𝑣₂(𝑧)=𝑣₁(𝑧), (89) with 𝑤(𝑧)=√1−𝑧2√𝑐𝑧,𝑎₁(𝑧)=1√2𝑤(𝑧)²(𝑧),𝑏+4+𝑤₁(𝑧)=1√2𝑤(𝑧)²+4−𝑤(𝑧). (90) For 𝑁=3, the roots explicitly write as 𝑢₁(𝑧)=122−𝑤(𝑧)−𝑎₁(𝑧)−𝑏₁(𝑧)√3𝑤(𝑧−𝑖)2𝑤(𝑧)√3−𝑎₁(𝑧)√3+𝑏₁(𝑧)𝑤(𝑧,𝑢₂)(𝑧)=1−𝑤(𝑧)−𝑎₂(𝑧)𝑤(𝑧),𝑢₃(𝑧)=𝑢₁(𝑧),𝑣₁(𝑧)=1𝑢₁(𝑧),𝑣₂(𝑧)=1𝑢₂(𝑧),𝑣₃(𝑧)=𝑣₁(𝑧), (91) with 𝑎₁(𝑧)=1√2𝑤(𝑧)²(𝑧)(𝑧)𝑏−2𝑤+4+𝑤−1,₁(𝑧)=1√2𝑤(𝑧)²(𝑧)(𝑧)𝑎−2𝑤+4−𝑤+1,₂(𝑧)=𝑤(𝑧)+2,𝑤(𝑧)=³√1−𝑧2√³.𝑐𝑧 (92)
Now, 𝐺_𝑘(𝑧) can be evaluated by residues theorem. Suppose first that 𝑘≥0 (then 𝑘+𝑁−1≥0) so that 0 is not a pole in the integral defining 𝐺_𝑘(𝑧): 𝐺_𝑘(𝑧)=^𝑁_𝑗=1Res𝜁^{𝑘+𝑁−1}𝑃(𝑧,𝜁),𝜁=𝑢_𝑗(𝑧)=^𝑁_𝑗=1𝑢_𝑗(𝑧)^{𝑘+𝑁−1}(𝜕𝑃/𝜕𝜁)𝑧,𝑢_𝑗(𝑧)=1𝑁(1−𝑧)^𝑁_𝑗=11−𝑢_𝑗(𝑧)1+𝑢_𝑗(𝑧)𝑢_𝑗(𝑧)^𝑘. (93) The foregoing representation of 𝐺_𝑘(𝑧) is valid a priori for any 𝑧∈(0,1/𝑀_∞). Actually, in view of the expressions of 𝑤(𝑧) and 𝑢_𝑗(𝑧), we can see that (93) defines an analytical function in the interval (0,1). Since 𝐺_𝑘(𝑧) is a power series, by analytical continuation, equality (93) holds true for any 𝑧∈(0,1). Moreover, by symmetry, we have that 𝐺_𝑘(𝑧)=𝐺_−𝑘(𝑧) for 𝑘≤0. We display this result in the theorem below.

Theorem 11. For any 𝑘∈ℤ, the generating function of the ℙ{𝑆_𝑛=𝑘}, 𝑛∈ℕ, is given, for any 𝑧∈(0,1), by 𝐺_𝑘(𝑧)=1𝑁(1−𝑧)^𝑁_𝑗=11−𝑢_𝑗(𝑧)1+𝑢_𝑗(𝑧)𝑢_𝑗(𝑧)^|𝑘|. (94)

Remark 12. Another proof of Theorem 11 consists in expanding the rational fraction 𝜁↦𝐺(𝑧,𝜁) into partial fractions. We find it interesting to outline the main steps of this method. We can write that 𝐺(𝑧,𝜁)=^𝑁_𝑗=1𝑢_𝑗(𝑧)𝜁−𝑢_𝑗(𝑧)+^𝑁_𝑗=1𝑣_𝑗(𝑧)𝜁−𝑣_𝑗(𝑧) (95) with 𝑢_𝑗(𝑧)=𝑢_𝑗(𝑧)𝑁(1−𝑧)1−𝑢_𝑗(𝑧)1+𝑢_𝑗(𝑧),𝑣_𝑗(𝑧)=𝑣_𝑗(𝑧)𝑁(1−𝑧)1−𝑣_𝑗(𝑧)1+𝑣_𝑗(𝑧)=−1/𝑢_𝑗(𝑧)𝑁(1−𝑧)1−𝑢_𝑗(𝑧)1+𝑢_𝑗(𝑧). (96) We next expand the partial fractions 1/(𝜁−𝑢_𝑗(𝑧)) and 1/(𝜁−𝑣_𝑗(𝑧)) into power series as follows. We have checked that |𝑢_𝑗(𝑧)|<1<|𝑣_𝑗(𝑧)| for any 𝑗∈{1,…,𝑁}. Now, if |𝑢_𝑗(𝑧)|<|𝜁|<|𝑣_𝑗(𝑧)| for any 𝑗∈{1,…,𝑁}, 1𝜁−𝑢_𝑗(𝑧)=^∞_𝑘=0𝑢_𝑗(𝑧)^𝑘𝜁^𝑘+1=⁻¹_𝑘=−∞𝜁^𝑘𝑢_𝑗(𝑧)^𝑘+1,1𝜁−𝑣_𝑗(𝑧)=−^∞_𝑘=0𝜁^𝑘𝑣_𝑗(𝑧)^𝑘+1 (97) from which (94) can be easily extracted.

2.3. Limiting Pseudoprocess

In this section, by pseudoprocess it is meant a continuous-time process driven by a signed measure. Actually, this object is not properly defined on all continuous times but only on dyadic times 𝑘/2^𝑗, 𝑗,𝑘∈ℕ. A proper definition consists in seeing it as the limit of a step process associated with the observations of the pseudoprocess on the dyadic times. We refer the reader to [10, 18] for precise details which are cumbersome to reproduce here.

Below, we give an ad hoc definition for the convergence of a family of pseudoprocesses ((𝑋^𝜀_𝑡)_𝑡≥0)_𝜀>0 towards a pseudoprocess (𝑋_𝑡)_𝑡≥0.

Definition 13. Let ((𝑋^𝜀_𝑡)_𝑡≥0)_𝜀>0 be a family of pseudoprocesses and (𝑋_𝑡)_𝑡≥0 a pseudoprocess. We say that 𝑋^𝜀_𝑡𝑡≥0⟶_𝜀→0+𝑋_𝑡𝑡≥0 (98) if and only if ∀𝑛∈ℕ^∗,∀𝑡₁,…,𝑡_𝑛≥0,∀𝜇₁,…,𝜇_𝑛𝔼∈ℝ,e^{𝑖∑𝑛𝑘=1𝜇𝑘𝑋𝜀𝑡𝑘}⟶_𝜀→0+𝔼e^{𝑖∑𝑛𝑘=1𝜇𝑘𝑋𝑡𝑘}. (99)

This is the weak convergence of the finite-dimensional projections of the family of pseudoprocesses.

In this part, we choose for the family ((𝑋^𝜀_𝑡)_𝑡≥0)_𝜀>0 the continuous-time pseudoprocesses defined, for any 𝜀>0, by 𝑋^𝜀_𝑡=𝜀𝑆_{⌊𝑡/𝜀2𝑁⌋},𝑡≥0, (100) where ⌊⋅⌋ stands for the usual floor function. The quantity 𝑋^𝜀_𝑡 takes its values on the discrete set 𝜀ℤ. Roughly speaking, we normalize the pseudorandom walk on the time × space grid 𝜀^2𝑁ℕ×𝜀ℤ. Let (𝑋_𝑡)_𝑡≥0 be the pseudo-Brownian motion. It is characterized by the following property: for any 𝑛∈ℕ^∗, any 𝑡₁,…,𝑡_𝑛≥0 such that 𝑡₁<⋯<𝑡_𝑛 and any 𝜇₁,…,𝜇_𝑛∈ℝ, 𝔼e^{𝑖∑𝑛𝑘=1𝜇𝑘𝑋𝑡𝑘}=e^{−𝑐∑𝑛𝑘=1(𝜇1+⋯+𝜇𝑘)2𝑁(𝑡𝑘−𝑡𝑘−1)}. (101) We refer to [10, 18] for a proper definition of pseudo-Brownian motion, and to references therein for interesting properties of this pseudoprocess.

Theorem 14. Suppose that 𝑐≤1/2^2𝑁−1. The following convergence holds: 𝑋^𝜀_𝑡𝑡≥0⟶_𝜀→0+𝑋_𝑡𝑡≥0. (102)

Proof. (i) We begin by computing the Laplace-Fourier transform of 𝑋^𝜀_𝑡. By definition of 𝑋^𝜀_𝑡, we have that 𝔼(e^{𝑖𝜇𝑋𝜀𝑡})=𝔼(e^{𝑖𝜇𝜀𝑆⌋⌊𝑡/𝜀2𝑁}) and then ₀^+∞e^−𝜆𝑡𝔼e^{𝑖𝜇𝑋𝜀𝑡}d𝑡=^∞_𝑛=0^{(𝑛+1)𝜀2𝑁}_{𝑛𝜀2𝑁}e^−𝜆𝑡d𝑡𝔼e^{𝑖𝜇𝜀𝑆𝑛}=1−e^{−𝜆𝜀2𝑁}𝜆^∞_𝑛=0e^{−𝜆𝜀2𝑁𝑛}𝔼e^{𝑖𝜇𝜀𝑆𝑛}=1−e^{−𝜆𝜀2𝑁}𝜆_{𝑛∈ℕ,𝑘∈ℤ}e^{−𝜆𝜀2𝑁𝑛}e^{𝑖𝜇𝜀𝑘}ℙ𝑆_𝑛==𝑘1−e^{−𝜆𝜀2𝑁}𝜆𝐺e^{−𝜆𝜀2𝑁},e^𝑖𝜇𝜀. (103) By (71), we have that 𝐺e^{−𝜆𝜀2𝑁},e^𝑖𝜇𝜀=11−e^{−𝜆𝜀2𝑁}+𝑐4^𝑁e^{−𝜆𝜀2𝑁}sin^2𝑁.𝜇𝜀/2 (104)
Actually, equality (104) is valid for 𝜆 such that e^{−𝜆𝜀2𝑁}<1/𝑀_∞; that is, 𝜆>(ln𝑀_∞)/𝜀^2𝑁. Since 𝑐 is assumed not to be greater than 1/2^2𝑁−1, by (42), we have that 𝑀_∞=1 and (104) is valid for any 𝜆>0.
Now, by using the elementary asymptotics sin(𝜇𝜀/2)=_𝜀→0+𝜇𝜀/2+𝑜(𝜀) and e^{−𝜆𝜀2𝑁}=_𝜀→0+1−𝜆𝜀^2𝑁+𝑜(𝜀), we obtain that 𝐺e^{−𝜆𝜀2𝑁},e^𝑖𝜇𝜀∼_𝜀→0+1𝜆+𝑐𝜇^2𝑁1𝜀^2𝑁. (105) As a result, for any 𝜆>0, lim_𝜀→0+0^+∞e^−𝜆𝑡𝔼e^{𝑖𝜇𝑋𝜀𝑡}d1𝑡=𝜆+𝑐𝜇^2𝑁=₀^+∞e^{−(𝜆+𝑐𝜇2𝑁)𝑡}d𝑡 (106) from which and (101) we deduce that lim_𝜀→0+𝔼e^{𝑖𝜇𝑋𝜀𝑡}=e^{−𝑐𝜇2𝑁𝑡}=𝔼e^{𝑖𝜇𝑋𝑡}. (107) Notice that the Laplace-Fourier of 𝑋_𝑡 takes the simple form ₀^+∞e^−𝜆𝑡𝔼e^{𝑖𝜇𝑋𝑡}d1𝑡=𝜆+𝑐𝜇^2𝑁. (108)
(ii) In the same way, we compute the Laplace-Fourier transform of 𝑋^𝜀_{𝑡+𝜀2𝑛} which will be used further. We have 𝔼(e^{𝑖𝜇𝑋𝜀𝑡+𝜀2𝑛})=𝔼(e^{𝑖𝜇𝜀𝑆⌊𝑡/𝜀2𝑁⌋+1}). Then ₀^+∞e^−𝜆𝑡𝔼e^{𝑖𝜇𝑋𝜀𝑡+𝜀2𝑛}d𝑡=^∞_𝑛=0^{(𝑛+1)𝜀2𝑁}_{𝑛𝜀2𝑁}e^−𝜆𝑡d𝑡𝔼e^{𝑖𝜇𝜀𝑆𝑛+1}=1−e^{−𝜆𝜀2𝑁}𝜆^∞_𝑛=0e^{−𝜆𝜀2𝑁𝑛}𝔼e^{𝑖𝜇𝜀𝑆𝑛+1}=1−e^{−𝜆𝜀2𝑁}𝜆e^{𝜆𝜀2𝑁}_{𝑛∈ℕ∗,𝑘∈ℤ}e^{−𝜆𝜀2𝑁𝑛}e^{𝑖𝜇𝜀𝑘}ℙ𝑆_𝑛==𝑘e^{𝜆𝜀2𝑁}−1𝜆𝐺e^{−𝜆𝜀2𝑁},e^𝑖𝜇𝜀−1. (109)
As for (107), we immediately extract the following limit: lim_𝜀→0+𝔼e^{𝑖𝜇𝑋𝜀𝑡+𝜀2𝑁}=𝔼e^{𝑖𝜇𝑋𝑡}. (110)
(iii) We now compute the joint Fourier transform of (𝑋^𝜀_𝑡1,𝑋^𝜀_𝑡2) for two times 𝑡₁, 𝑡₂ such that 𝑡₁<𝑡₂. Using the elementary fact that ⌊𝑥⌋−⌊𝑦⌋∈{⌊𝑥−𝑦⌋,⌊𝑥−𝑦⌋+1}, we observe that 𝑋^𝜀_𝑡2−𝑋=^𝜀_𝑡1^𝑑𝜀𝑆_{⌊𝑡2/𝜀2𝑁⌋−⌊𝑡1/𝜀2𝑁⌋}∈𝜀𝑆_{⌊(𝑡2−𝑡1)/𝜀2𝑁⌋},𝜀𝑆_{⌊(𝑡2−𝑡1)/𝜀2𝑁⌋+1}=𝑋^𝜀_{𝑡2−𝑡1},𝑋^𝜀_{𝑡2−𝑡1+𝜀2𝑁}. (111) Then, we get, for 𝜇₁,𝜇₂∈ℝ, that 𝔼e^{𝑖(𝜇1𝑋𝜀𝑡1+𝜇2𝑋𝜀𝑡2)}∈𝔼e^{𝑖(𝜇1+𝜇2)𝑋𝜀𝑡1}𝔼e^{𝑖𝜇2𝑋𝜀𝑡21−𝑡},𝔼e^{𝑖(𝜇1+𝜇2)𝑋𝜀𝑡1}𝔼e^{𝑖𝜇2𝑋𝜀𝑡21−𝑡+𝜀2𝑁}. (112) By (107) and (110), we obtain the following limit: lim_𝜀→0+𝔼e^{𝑖𝜇2𝑋𝜀𝑡21−𝑡}=lim_𝜀→0+𝔼e^{𝑖𝜇2𝑋𝜀𝑡21−𝑡+𝜀2𝑁}=𝔼e^{𝑖𝜇2𝑋𝑡21−𝑡} (113) which yields that lim_𝜀→0+𝔼e^{𝑖(𝜇1𝑋𝜀𝑡1+𝜇2𝑋𝜀𝑡2)}=lim_𝜀→0+𝔼e^{𝑖(𝜇1+𝜇2)𝑋𝜀𝑡1}×lim_𝜀→0+𝔼e^{𝑖𝜇2𝑋𝜀𝑡21−𝑡}=𝔼e^{𝑖(𝜇1+𝜇2)𝑋𝑡1}×𝔼e^{𝑖𝜇2𝑋𝑡21−𝑡}=𝔼e^{𝑖(𝜇1𝑋𝑡1+𝜇2𝑋𝑡2)}. (114)
(iv) Finally, we can easily extend the foregoing limiting result by recurrence as follows: for 𝑛∈ℕ^∗, 𝜇₁,…,𝜇_𝑛∈ℝ and for any times 𝑡₁,…,𝑡_𝑛 such that 𝑡₁<⋯<𝑡_𝑛, lim_𝜀→0+𝔼e^{𝑖(𝜇1𝑋𝜀𝑡1+⋯+𝜇𝑛𝑋𝜀𝑡𝑛)}=𝔼e^{𝑖(𝜇1𝑋𝑡1+⋯+𝜇𝑛𝑋𝑡𝑛)}. (115)
The proof of Theorem 14 is complete.

We find it interesting to compute in a similar way the 𝜆-potential of the pseudoprocess (𝑋_𝑡)_𝑡≥0. By definition of 𝑋^𝜀_𝑡, we have, for any 𝛼,𝛽∈ℝ such that 𝛼<𝛽, ℙ{𝑋^𝜀_𝑡∈[𝛼,𝛽)}=ℙ{𝑆_{⌊𝑡/𝜀2𝑁⌋}∈[𝛼/𝜀,𝛽/𝜀)}. Thus, ₀^+∞e^−𝜆𝑡ℙ𝑋^𝜀_𝑡∈𝛼,𝛽d𝑡=^∞_𝑛=0^{(𝑛+1)𝜀2𝑁}_{𝑛𝜀2𝑁}e^−𝜆𝑡d𝑡ℙ𝑆_𝑛∈𝛼𝜀,𝛽𝜀=1−e^{−𝜆𝜀2𝑁}𝜆^∞_𝑛=0e^{−𝜆𝜀2𝑁𝑛}_{𝑘∈ℤ∶𝛼/𝜀≤𝑘<𝛽/𝜀}ℙ𝑆_𝑛==𝑘1−e^{−𝜆𝜀2𝑁}𝜆_{𝑘∈ℤ∶𝛼/𝜀≤𝑘<𝛽/𝜀}^∞_𝑛=0ℙ𝑆_𝑛=𝑘e^{−𝜆𝜀2𝑁𝑛}=1−e^{−𝜆𝜀2𝑁}𝜆_{𝑘∈ℤ∶𝛼/𝜀≤𝑘<𝛽/𝜀}𝐺_𝑘e^{−𝜆𝜀2𝑁}. (116)

Interchanging the two sums in the above computations is justified by the fact that the series ∑^∞_𝑛=0ℙ{𝑆_𝑛=𝑘}(e^{−𝜆𝜀2𝑁})^𝑛 is absolutely convergent because of the condition 𝑐≤1/2^2𝑁−1. Indeed, by (53), for any 𝜆>0, |ℙ{𝑆_𝑛=𝑘}|<𝑀^𝑛_∞=1.

Put 𝑢_𝑗(𝜆,𝜀)=𝑢_𝑗(e^{−𝜆𝜀2𝑁}). This yields that ₀^+∞e^−𝜆𝑡ℙ𝑋^𝜀_𝑡∈𝛼,𝛽d𝑡=1𝑁𝜆^𝑁_𝑗=11−𝑢_𝑗(𝜆,𝜀)1+𝑢_𝑗(𝜆,𝜀)_{𝛼/𝜀≤𝑘<𝛽/𝜀}𝑢_𝑗(𝜆,𝜀)^|𝑘|. (117) Suppose, for example, that 0≤𝛼<𝛽. Then, _{𝛼/𝜀≤𝑘<𝛽/𝜀}𝑢_𝑗(𝜆,𝜀)^|𝑘|=^{⌈𝛽/𝜀⌉−1}_{⌈⌉𝑘=𝛼/𝜀}𝑢_𝑗(𝜆,𝜀)^𝑘=𝑢_𝑗(𝜆,𝜀)^{⌈𝛼/𝜀⌉}−𝑢_𝑗(𝜆,𝜀)^{⌈𝛽/𝜀⌉}1−𝑢_𝑗(𝜆,𝜀) (118) where ⌈⋅⌉ stands for the usual ceiling function. By using (85), we deduce that 𝑢_𝑗(𝜆,𝜀)=_𝜀→0+1−𝜑_𝑗^2𝑁𝜆𝑐𝜀+𝑜(𝜀) (119) which implies that lim_𝜀→0+𝑢_𝑗(𝜆,𝜀)^{⌈𝛼/𝜀⌉}=e^{−𝜑𝑗2𝑁𝜆/𝑐𝛼}. (120) Therefore, lim_𝜀→0+0^+∞e^−𝜆𝑡ℙ𝑋^𝜀_𝑡∈𝛼,𝛽d𝑡=12𝑁𝜆^𝑁_𝑗=1e^{−𝜑𝑗2𝑁𝜆/𝑐𝛼}−e^{−𝜑𝑗2𝑁𝜆/𝑐𝛽}=12𝑁𝜆^𝑁_𝑗=1𝜑_𝑗^2𝑁𝜆𝑐^𝛽_𝛼e^{−𝜑𝑗2𝑁𝜆/𝑐𝑥}d𝑥. (121) The case 𝛼<𝛽≤0 is similar to treat. We have obtained the following result.

Proposition 15. The 𝜆-potential of the pseudoprocess (𝑋_𝑡)_𝑡≥0 is given by ₀^+∞e^−𝜆𝑡ℙ𝑋_𝑡∈d𝑥d𝑥d𝑡=12𝑁^2𝑁√𝑐𝜆^{𝑁1−1/(2𝑁)}_𝑗∑₌₁𝜑_𝑗e^{−𝜑𝑗2𝑁𝜆/𝑐𝑥}−1𝑖𝑓𝑥≥0,2𝑁^2𝑁√𝑐𝜆^{𝑁1−1/(2𝑁)}_𝑗∑₌₁𝜑_𝑗e^{𝜑𝑗2𝑁𝜆/𝑐𝑥}𝑖𝑓𝑥≤0. (122)

3. Part II—First Overshooting Time of a Single Threshold

3.1. On the Pseudodistribution of (𝜎⁺_𝑏,𝑆⁺_𝑏)

Let 𝑏∈ℕ^∗. In this section, we explicitly compute the generating function of (𝜎⁺_𝑏,𝑆⁺_𝑏). Set, for ℓ∈{𝑏,𝑏+1,…,𝑏+𝑁−1}, 𝐻⁺_𝑏,ℓ(𝑧)=𝔼𝑧^𝜎+𝑏𝟙_{𝑆⁺_{𝑏=ℓ,𝜎}⁺_𝑏<+∞}=_𝑘∈ℕℙ𝜎⁺_𝑏=𝑘,𝑆⁺_𝑏=ℓ𝑧^𝑘. (123) We are able to provide an explicit expression of 𝐻⁺_𝑏,ℓ(𝑧). Before tackling this problem, we need an a priori estimate for ℙ{𝜎⁺_𝑏=𝑘,𝑆⁺_𝑏=ℓ}. By (62), we immediately derive that |ℙ{𝜎⁺_𝑏=𝑘,𝑆⁺_𝑏=ℓ}|=|ℙ{𝑆₁<𝑏,…,𝑆_𝑘−1<𝑏,𝑆_𝑘=ℓ}|≤𝑀^𝑘₁. Hence, the power series defining 𝐻⁺_𝑏,ℓ(𝑧) absolutely converges for |𝑧|<1/𝑀₁.

3.1.1. Joint Pseudodistribution of (𝜎⁺_𝑏,𝑆⁺_𝑏)

Theorem 16. The pseudodistribution of (𝜎⁺_𝑏,𝑆⁺_𝑏) is characterized by the identity, valid for any 𝑧∈(0,1) and any ℓ∈{𝑏,𝑏+1,…,𝑏+𝑁−1}, 𝔼𝑧^𝜎+𝑏𝟙_{𝑆⁺_{𝑏=ℓ,𝜎}⁺_𝑏<+∞}=(−1)^{𝑁ℓ−𝑏}_𝑘=1𝑠⁺_{𝑘,ℓ−𝑏}(𝑧)𝑝⁺_𝑘(𝑧)𝑢_𝑘(𝑧)^{𝑏+𝑁−1}, (124) where 𝑠⁺_𝑘,0(𝑧)=1 and for 𝑘∈{1,…,𝑁}, ℓ∈{1,…,𝑁−1}, 𝑠⁺_𝑘,ℓ(𝑧)=_{1≤𝑖1<⋯<𝑖ℓ𝑖≤𝑁1,…,𝑖ℓ≠𝑘}𝑢_𝑖1(𝑧)⋯𝑢_𝑖ℓ(𝑧),𝑝⁺_𝑘(𝑧)=_{1≤𝑗≤𝑁𝑗≠𝑘}𝑢_𝑘(𝑧)−𝑢_𝑗(𝑧). (125)

Proof. Pick an integer 𝑘≥𝑏. If 𝑆_𝑛=𝑘, then an overshoot of the threshold 𝑏 occurs before time 𝑛: 𝜎⁺_𝑏≤𝑛. This remark and the independence of the increments of the pseudorandom walk entail that ℙ𝑆_𝑛=𝑘=ℙ𝑆_𝑛=𝑘,𝜎⁺_𝑏=≤𝑛_{𝑛𝑗=0𝑏+𝑁−1ℓ=𝑏}ℙ𝑆_𝑛=𝑘,𝜎⁺_𝑏=𝑗,𝑆⁺_𝑏==ℓ_{𝑛𝑗=0𝑏+𝑁−1ℓ=𝑏}ℙ𝜎⁺_𝑏=𝑗,𝑆⁺_𝑏=ℓℙ𝑆_𝑛−𝑗=𝑘−ℓ. (126) Since the series defining 𝐺_𝑘(𝑧) and 𝐻⁺_𝑏,ℓ(𝑧) absolutely converge, respectively, for 𝑧∈(0,1) and |𝑧|<1/𝑀₁, and since 𝑀₁≥1, we can apply the generating function to the convolution equality (126). We get, for 𝑧∈(0,1/𝑀₁), that 𝐺_𝑘(𝑧)=^{𝑏+𝑁−1}_ℓ=𝑏𝐺_𝑘−ℓ(𝑧)𝐻⁺_𝑏,ℓ(𝑧). (127) Using expression (94) of 𝐺_𝑘, namely, 𝐺_𝑘(𝑧)=∑^𝑁_𝑗=1𝛼_𝑗(𝑧)𝑢_𝑗(𝑧)^𝑘 for 𝑘≥0, where 𝛼_𝑗(𝑧)=1/[𝑁(1−𝑧)]×[1−𝑢_𝑗(𝑧)]/[1+𝑢_𝑗(𝑧)], we obtain that ^𝑁_𝑗=1𝛼_𝑗(𝑧)𝑢_𝑗(𝑧)^{𝑘𝑏+𝑁−1}_ℓ=𝑏𝐻⁺_𝑏,ℓ(𝑧)𝑢_𝑗(𝑧)^ℓ−1=0,𝑘≥𝑏+𝑁−1. (128) Recalling that 𝑣_𝑗(𝑧)=1/𝑢_𝑗(𝑧) and setting 𝛼_𝑗(𝑧)=𝛼_𝑗(𝑧)(∑^{𝑏+𝑁−1}_ℓ=𝑏𝐻⁺_𝑏,ℓ(𝑧)𝑣_𝑗(𝑧)^ℓ−1), system (128) reads ∑^𝑁_𝑗=1𝛼_𝑗(𝑧)𝑢_𝑗(𝑧)^𝑘=0, 𝑘≥𝑏+𝑁−1. When limiting the range of 𝑘 to the set {𝑏+𝑁,𝑏+𝑁+1,…,𝑏+2𝑁−1}, this becomes a homogeneous Vandermonde system whose solution is trivial: 𝛼_𝑗(𝑧)=0, 1≤𝑗≤𝑁. Thus, we get the following Vandermonde system: ^{𝑏+𝑁−1}_ℓ=𝑏𝐻⁺_𝑏,ℓ(𝑧)𝑣_𝑗(𝑧)^ℓ=1,1≤𝑗≤𝑁. (129) System (129) can be explicitly solved. In order to simplify the settings, we will omit the variable 𝑧 in the sequel of the proof. It is convenient to rewrite (129) as ^{𝑏+𝑁−1}_ℓ=𝑏𝐻⁺_𝑏,ℓ𝑣_𝑗^ℓ−𝑏=𝑢^𝑏_𝑗,1≤𝑗≤𝑁. (130) Cramer’s formulae yield 𝐻⁺_𝑏,ℓ=𝑉_ℓ𝑣₁,…,𝑣_𝑁𝑉𝑣₁,…,𝑣_𝑁,𝑏≤ℓ≤𝑏+𝑁−1, (131) where 𝑉𝑣₁,…,𝑣_𝑁=1𝑣₁⋯𝑣₁^𝑁−11𝑣₂⋯𝑣₂^𝑁−1⋮⋮⋮1𝑣_𝑁⋯𝑣_𝑁^𝑁−1=_{1≤𝑖<𝑗≤𝑁}𝑣_𝑗−𝑣_𝑖 (132) and, for any ℓ∈{𝑏,…,𝑏+𝑁−1}, 𝑉_ℓ𝑣₁,…,𝑣_𝑁=1𝑣₁⋯𝑣₁^{ℓ−𝑏−1}𝑢^𝑏₁𝑣₁^ℓ−𝑏+1⋯𝑣₁^𝑁−11𝑣₂⋯𝑣₂^{ℓ−𝑏−1}𝑢^𝑏₂𝑣₂^ℓ−𝑏+1⋯𝑣₂^𝑁−1⋮⋮⋮⋮⋮⋮1𝑣_𝑁⋯𝑣_𝑁^{ℓ−𝑏−1}𝑢^𝑏_𝑁𝑣_𝑁^ℓ−𝑏+1⋯𝑣_𝑁^𝑁−1. (133) This last determinant can be expanded as ∑^𝑁_𝑘=1𝑢^𝑏_𝑘𝑉_𝑘ℓ(𝑣₁,…,𝑣_𝑘−1,𝑣_𝑘+1,…,𝑣_𝑁) with, for 𝑘∈{1,…,𝑁}, 𝑉_𝑘ℓ𝑣₁,…,𝑣_𝑘−1,𝑣_𝑘+1,…,𝑣_𝑁=1𝑣₁⋯𝑣₁^{ℓ−𝑏−1}0𝑣₁^ℓ−𝑏+1⋯𝑣₁^𝑁−1⋮⋮⋮⋮⋮⋮1𝑣_𝑘−1⋯𝑣^{ℓ−𝑏−1}_𝑘−10𝑣^ℓ−𝑏+1_𝑘−1⋯𝑣^𝑁−1_𝑘−11𝑣_𝑘⋯𝑣_𝑘^{ℓ−𝑏−1}1𝑣_𝑘^ℓ−𝑏+1⋯𝑣_𝑘^𝑁−11𝑣_𝑘+1⋯𝑣^{ℓ−𝑏−1}_𝑘+10𝑣^ℓ−𝑏+1_𝑘+1⋯𝑣^𝑁−1_𝑘+1⋮⋮⋮⋮⋮⋮1𝑣_𝑁⋯𝑣_𝑁^{ℓ−𝑏−1}0𝑣_𝑁^ℓ−𝑏+1⋯𝑣_𝑁^𝑁−1. (134) In fact, the quantity 𝑉_𝑘ℓ(𝑣₁,…,𝑣_𝑘−1,𝑣_𝑘+1,…,𝑣_𝑁) is the coefficient of 𝑥^ℓ−𝑏 in the polynomial 𝑥⟼1𝑣₁⋯𝑣₁^𝑁−1⋮⋮⋮1𝑣_𝑘−1⋯𝑣^𝑁−1_𝑘−11𝑥⋯𝑥^𝑁−11𝑣_𝑘+1⋯𝑣^𝑁−1_𝑘+1⋮⋮⋮1𝑣_𝑁⋯𝑣_𝑁^𝑁−1 (135) which is nothing but 𝑉(𝑣₁,…,𝑣_𝑘−1,𝑥,𝑣_𝑘+1,…,𝑣_𝑁), the value of which is _{1≤𝑖<𝑗≤𝑁𝑖,𝑗≠𝑘}𝑣_𝑗−𝑣_{𝑖1≤𝑖≤𝑘−1}𝑥−𝑣_{𝑖𝑘+1≤𝑖≤𝑁}𝑣_𝑖=∏−𝑥_{1≤𝑖<𝑗≤𝑁}𝑣_𝑗−𝑣_𝑖∏_{1≤𝑖≤𝑁𝑖≠𝑘}𝑣_𝑘−𝑣_{𝑖1≤𝑖≤𝑁𝑖≠𝑘}𝑥−𝑣_𝑖=𝑉𝑣₁,…,𝑣_{𝑁1≤𝑖≤𝑁𝑖≠𝑘}𝑥−𝑣_𝑖𝑣_𝑘−𝑣_𝑖=(−1)^𝑁−1𝑉𝑣₁,…,𝑣_{𝑁1≤𝑖≤𝑁𝑖≠𝑘}𝑢_𝑘𝑢_𝑖𝑥−1𝑢_𝑘−𝑢_𝑖=(−1)^𝑁−1𝑢_𝑘^𝑁−1𝑝⁺_𝑘𝑉𝑣₁,…,𝑣_{𝑁1≤𝑖≤𝑁𝑖≠𝑘}𝑢_𝑖𝑥−1. (136) Using the elementary expansion ∏_{1≤𝑖≤𝑁𝑖≠𝑘}(𝑢_𝑖𝑥−1)=∑^𝑁−1_ℓ=0(−1)^{𝑁−1−ℓ}𝑠⁺_𝑘,ℓ𝑥^ℓ, we obtain by identification that 𝑉_𝑘ℓ𝑣₁,…,𝑣_𝑘−1,𝑣_𝑘+1,…,𝑣_𝑁=(−1)^ℓ−𝑏𝑢_𝑘^𝑁−1𝑝⁺_𝑘𝑠⁺_{𝑘,ℓ−𝑏}𝑉𝑣₁,…,𝑣_𝑁. (137) Plugging this expression into (131), we then derive for 𝐻⁺_𝑏,ℓ(𝑧) representation (124) which is valid at least for 𝑧∈(0,1/𝑀₁). Finally, we observe that (124) defines an analytical function in (0,1) and that 𝐻⁺_𝑏,ℓ(𝑧) is a power series. Thus, by analytical continuation, (124) holds true for any 𝑧∈(0,1).

Example 17. For 𝑁=1, the settings of Theorem 16 write 𝑠⁺_1,0(𝑧)=𝑝⁺₁(𝑧)=1. Then, formula (124) reads 𝔼𝑧^𝜎+𝑏𝟙_{𝑆⁺_{𝑏=𝑏,𝜎}⁺_𝑏<+∞}=𝑢₁(𝑧)^𝑏, (138) where 𝑢₁(𝑧) is given in Example 10. Of course, in this case, the condition 𝑆⁺_𝑏=𝑏 is redundant since we are dealing with an ordinary random walk with jumps of one unity at most. When 𝑐=1/2, this is the classical symmetric random walk and (124) recovers the most well-known formula in random walk theory: 𝔼𝑧^𝜎+𝑏𝟙_{𝜎⁺_𝑏<+∞}=1−√1−𝑧²𝑧^𝑏. (139) For 𝑁=2, the settings of Theorem 16 write 𝑠⁺_1,0(𝑧)=𝑠⁺_2,0(𝑧)=1,𝑠⁺_1,1(𝑧)=𝑢₂(𝑧),𝑠⁺_2,1(𝑧)=𝑢₁(𝑧),𝑝⁺₁(𝑧)=𝑢₁(𝑧)−𝑢₂(𝑧),𝑝⁺₂(𝑧)=𝑢₂(𝑧)−𝑢₁(𝑧), (140) where 𝑢₁(𝑧) and 𝑢₂(𝑧) are given in Example 10 and (124) reads 𝔼𝑧^𝜎+𝑏𝟙_{𝑆⁺_{𝑏=𝑏,𝜎}⁺_𝑏<+∞}=𝑢₁(𝑧)^𝑏+1−𝑢₂(𝑧)^𝑏+1𝑢₁(𝑧)−𝑢₂(𝑧),𝔼𝑧^𝜎+𝑏𝟙_{𝑆⁺_{𝑏=𝑏+1,𝜎}⁺_𝑏<+∞}=𝑢₁(𝑧)𝑢₂(𝑧)^𝑏+1−𝑢₂(𝑧)𝑢₁(𝑧)^𝑏+1𝑢₁(𝑧)−𝑢₂(𝑧). (141)

Remark 18. We have the similar expression related to 𝜎⁻_𝑎 below. The analogous system to (129) writes as ^𝑎_{ℓ=𝑎−𝑁+1}𝐻⁻_𝑎,ℓ(𝑧)𝑢_𝑗(𝑧)^ℓ=1,1≤𝑗≤𝑁, (142) where 𝐻⁻_𝑎,ℓ(𝑧)=𝔼(𝑧^𝜎−𝑎𝟙_{𝑆⁻_{𝑎=ℓ,𝜎−𝑎<+∞}}). The solution is given by 𝔼𝑧^𝜎−𝑎𝟙_{𝑆⁻_{𝑎=ℓ,𝜎−𝑎<+∞}}=(−1)^{𝑁ℓ−𝑎}_𝑘=1𝑠⁻_{𝑘,ℓ−𝑎}(𝑧)𝑝⁻_𝑘(𝑧)𝑢_𝑘(𝑧)^{𝑎+𝑁−1}, (143) where 𝑠⁻_𝑘,0(𝑧)=1 and, for 𝑘∈{1,…,𝑁}, ℓ∈{1,…,𝑁−1}, 𝑠⁻_𝑘,ℓ(𝑧)=_{1≤𝑖1<⋯<𝑖ℓ𝑖≤𝑁1,…,𝑖ℓ≠𝑘}𝑣_𝑖1(𝑧)⋯𝑣_𝑖ℓ(𝑧),𝑝⁻_𝑘(𝑧)=_{1≤𝑗≤𝑁𝑗≠𝑘}𝑣_𝑘(𝑧)−𝑣_𝑗(𝑧). (144)(145)
The double generating function of (𝜎⁺_𝑏,𝑆⁺_𝑏) defined by 𝔼𝑧^𝜎+𝑏𝜁^𝑆+𝑏𝟙_{𝜎⁺_𝑏<+∞}=^{𝑏+𝑁−1}_ℓ=𝑏𝔼𝑧^𝜎+𝑏𝟙_{𝑆⁺_𝑏=ℓ}𝜁^ℓ (146) admits an interesting representation by means of Lagrange interpolating polynomials that we display in the theorem below.

Theorem 19. The double generating function of (𝜎⁺_𝑏,𝑆⁺_𝑏) is given, for any 𝑧∈(0,1/𝑀₁) and 𝜁∈ℂ, by 𝔼𝑧^𝜎+𝑏𝜁^𝑆+𝑏𝟙_{𝜎⁺_𝑏<+∞}=^𝑁_𝑘=1𝐿_𝑘(𝑧,𝜁)𝑢_𝑘(𝑧)𝜁^𝑏, (147) where 𝐿_𝑘(𝑧,𝜁)=_{1≤𝑗≤𝑁𝑗≠𝑘}𝜁−𝑣_𝑗(𝑧)𝑣_𝑘(𝑧)−𝑣_𝑗(𝑧),𝑘∈{1,…,𝑁}, (148) are the Lagrange interpolating polynomials with respect to the variable 𝜁 such that 𝐿_𝑘(𝑧,𝑣_𝑗(𝑧))=𝛿_𝑗𝑘.

Proof. By (131) and by omitting the variable 𝑧 as previously mentioned, we have that 𝔼𝑧^𝜎+𝑏𝜁^𝑆+𝑏𝟙_{𝜎⁺_𝑏<+∞}=^{𝑏+𝑁−1}_ℓ=𝑏𝐻⁺_𝑏,ℓ𝜁^ℓ=^{𝑏+𝑁−1}_ℓ=𝑏𝑉_ℓ𝑣₁,…,𝑣_𝑁𝑉𝑣₁,…,𝑣_𝑁𝜁^ℓ=^{𝑏+𝑁−1}_ℓ=𝑏^𝑁_𝑘=1𝑢^𝑏_𝑘𝑉_𝑘ℓ𝑣₁,…,𝑣_𝑘−1,𝑣_𝑘+1,…,𝑣_𝑁𝑉𝑣₁,…,𝑣_𝑁𝜁^ℓ=^𝑁_𝑘=1^{𝑏+𝑁−1}_ℓ=𝑏𝑉_𝑘ℓ𝑣₁,…,𝑣_𝑘−1,𝑣_𝑘+1,…,𝑣_𝑁𝑉𝑣₁,…,𝑣_𝑁𝜁^ℓ−𝑏𝑢_𝑘𝜁^𝑏=^𝑁_𝑘=1𝑉𝑣₁,…,𝑣_𝑘−1,𝜁,𝑣_𝑘+1,…,𝑣_𝑁𝑉𝑣₁,…,𝑣_𝑁𝑢_𝑘𝜁^𝑏. (149) It is clear that the quantity 𝑉(𝑣₁,…,𝑣_𝑘−1,𝜁,𝑣_𝑘+1,…,𝑣_𝑁)/𝑉(𝑣₁,…,𝑣_𝑁), which explicitly writes as ^{1𝑣1⋯𝑣1𝑁−1⋮⋮⋮1𝑣𝑘−1⋯𝑣𝑁−1𝑘−11𝜁⋯𝜁𝑁−11𝑣𝑘+1⋯𝑣𝑁−1𝑘+1⋮⋮⋮1𝑣𝑁⋯𝑣𝑁𝑁−1}_{1𝑣1⋯𝑣1𝑁−1⋮⋮⋮1𝑣𝑁⋯𝑣𝑁𝑁−1}, (150) defines a polynomial of the variable 𝜁 of degree 𝑁−1 which vanishes at 𝑣₁,…,𝑣_𝑘−1,𝑣_𝑘+1,…,𝑣_𝑁 and equals 1 at 𝑣_𝑘. Hence, by putting back the variable 𝑧, it coincides with the Lagrange polynomial 𝐿_𝑘(𝑧,𝜁) and formula (147) immediately ensues.

Example 20. For 𝑁=2, (147) reads 𝔼𝑧^𝜎+𝑏𝜁^𝑆+𝑏𝟙_{𝜎⁺_𝑏<+∞}=𝜁^𝑏𝑢₁(𝑧)^𝑏𝜁−𝑣₂(𝑧)𝑣₁(𝑧)−𝑣₂(𝑧)+𝑢₂(𝑧)^𝑏𝜁−𝑣₁(𝑧)𝑣₂(𝑧)−𝑣₁(𝑧)=𝜁^𝑏𝑢₁(𝑧)−𝑢₂(𝑧)𝑢₁(𝑧)^𝑏+1−𝑢₂(𝑧)^𝑏+1+𝑢₁(𝑧)𝑢₂(𝑧)^𝑏+1−𝑢₂(𝑧)𝑢₁(𝑧)^𝑏+1𝜁. (151) This is in good agreement with the formulae of Example 17. We retrieve a result of [21].

3.1.2. Pseudodistribution of 𝑆⁺_𝑏

In order to derive the pseudodistribution of 𝑆⁺_𝑏 which is characterized by the numbers 𝐻⁺_𝑏,ℓ(1), ℓ∈{𝑏,𝑏+1,…,𝑏+𝑁−1}, we solve the system obtained by taking the limit in (129) as 𝑧→1⁻.

Lemma 21. The following system holds: ^{𝑏+𝑁−1ℓ=𝑘+𝑏}𝑘𝐻ℓ−𝑏_+𝑏,ℓ(1)=(−1)^𝑘𝑏+𝑘−1𝑏−1,0≤𝑘≤𝑁−1. (152)

Proof. By (85), we have the expansion 𝑣_𝑗(𝑧)=1/𝑢_𝑗(𝑧)=1−𝜀_𝑗(𝑧), where 𝜀_𝑗(𝑧)=_𝑧→1−𝒪(^2𝑁√1−𝑧) for any 𝑗∈{1,…,𝑁}. Putting this into (129), we get that ^{𝑏+𝑁−1}_ℓ=𝑏1−𝜀_𝑗(𝑧)^ℓ−𝑏𝐻⁺_𝑏,ℓ(𝑧)=1−𝜀_𝑗(𝑧)^−𝑏; (153) that is, ^𝑁−1_𝑘=0(−1)^{𝑘𝑏+𝑁−1}_{ℓ=𝑏+𝑘}𝑘𝐻ℓ−𝑏_+𝑏,ℓ(𝑧)𝜀_𝑗(𝑧)^𝑘=^∞_𝑘=0(−1)^𝑘𝑘𝜀−𝑏_𝑗(𝑧)^𝑘=^∞_𝑘=0𝜀𝑏+𝑘−1𝑏−1^𝑗(𝑧)^𝑘. (154) Set 𝑀_𝑘(𝑧)=(−1)^{𝑘𝑏+𝑁−1}_{ℓ=𝑏+𝑘}𝑘𝐻ℓ−𝑏_+𝑏,ℓ(𝑧)−,𝑅𝑏+𝑘−1𝑏−1_𝑗(𝑧)=^∞_ℓ=𝑁𝜀𝑏+ℓ−1𝑏−1^𝑗(𝑧)^ℓ. (155) Then, equality (154) reads ^𝑁−1_𝑘=0𝑀_𝑘(𝑧)𝜀_𝑗(𝑧)^𝑘=𝑅_𝑗(𝑧),1≤𝑗≤𝑁. (156)
This is a Vandermonde system, the solution of which is given by 𝑀_𝑘(𝑧)=𝑉_𝑘(𝑧)/𝑉(𝑧), 0≤𝑘≤𝑁−1, where 𝑉(𝑧)=1𝜀₁(𝑧)⋯𝜀₁(𝑧)^𝑁−11𝜀₂(𝑧)⋯𝜀₂(𝑧)^𝑁−1⋮⋮⋮1𝜀_𝑁(𝑧)⋯𝜀_𝑁(𝑧)^𝑁−1,𝑉_𝑘(𝑧)=1𝜀₁(𝑧)⋯𝜀₁(𝑧)^𝑘−1𝑅₁(𝑧)𝜀₁(𝑧)^𝑘+1⋯𝜀₁(𝑧)^𝑁−11𝜀₂(𝑧)⋯𝜀₂(𝑧)^𝑘−1𝑅₂(𝑧)𝜀₂(𝑧)^𝑘+1⋯𝜀₂(𝑧)^𝑁−1⋮⋮⋮⋮⋮⋮1𝜀_𝑁(𝑧)⋯𝜀_𝑁(𝑧)^𝑘−1𝑅_𝑁(𝑧)𝜀_𝑁(𝑧)^𝑘+1⋯𝜀_𝑁(𝑧)^𝑁−1. (157) Since, by (85), 𝜀_𝑗(𝑧)∼_𝑧→1−constant×^2𝑁√1−𝑧 for any 𝑗∈{1,…,𝑁}, we have that 𝑉(𝑧)=_{1≤ℓ<𝑚≤𝑁}𝜀_𝑚(𝑧)−𝜀_ℓ(𝑧∼_𝑧→1−)constant×(1−𝑧)^(𝑁−1)/4 (158)
and second, 𝑅_𝑗(𝑧)∼_𝑧→1−𝜀𝑏+𝑁−1𝑏−1^𝑗(𝑧)^𝑁∼constant×√1−𝑧 (159)
which implies, for 𝑘∈{0,…,𝑁−1}, that 𝑉_𝑘(𝑧)=_𝑧→1−𝒪(1−𝑧)^{1/2+(∑1≤𝑚≤𝑁−1,𝑚≠𝑘𝑚)/(2𝑁)}=𝑜(1−𝑧)^(𝑁−1)/4. (160)
Therefore, lim_𝑧→1⁻𝑀_𝑘(𝑧)=0. On the other hand, for 𝑧∈(0,1), referring to the definition of 𝑀_𝑘(𝑧), we can see that the quantity 𝐻⁺_𝑏,ℓ(𝑧) can be expressed as a linear combination of the 𝑀_𝑘(𝑧)’s plus a constant. Hence, the limit lim_𝑧→1⁻𝐻⁺_𝑏,ℓ(𝑧) exists and, by appealing to a Tauberian theorem, it coincides with 𝐻⁺_𝑏,ℓ(1). This finishes the proof of (152).

Theorem 22. The pseudodistribution of 𝑆⁺_𝑏 is characterized by the following pseudo-probabilities: for any ℓ∈{𝑏,𝑏+1,…,𝑏+𝑁−1}, ℙ𝑆⁺_𝑏=ℓ,𝜎⁺_𝑏<+∞=(−1)^𝑏+ℓ𝑏ℓ𝑏.𝑁−1ℓ−𝑏𝑏+𝑁−1 (161) Moreover, ℙ{𝜎⁺_𝑏<+∞}=1.

Proof. We explicitly solve system (152) rewritten as ^𝑁−1_ℓ=𝑘ℓ𝑘𝐻⁺_{𝑏,ℓ+𝑏}(1)=(−1)^𝑘𝑏+𝑘−1𝑏−1,0≤𝑘≤𝑁−1. (162) The matrix of the system is (^ℓ_𝑘)_{0≤𝑘,ℓ≤𝑁−1} which admits ((−1)^𝑘+ℓℓ_𝑘)_{0≤𝑘,ℓ≤𝑁−1} as an inverse with the convention of settings ^ℓ_𝑘=0 if 𝑘>ℓ. The solution of the system is given, for ℓ∈{0,1,…,𝑁−1}, by 𝐻⁺_{𝑏,ℓ+𝑏}(1)=^𝑁−1_𝑘=ℓ(−1)^𝑘+ℓ𝑘ℓ×(−1)^𝑘=()𝑏+𝑘−1𝑏−1−1^ℓ𝑏+ℓ−1𝑏−1^{𝑁−1𝑘=ℓ}=()𝑏+𝑘−1𝑏+ℓ−1−1^ℓ=()𝑏+ℓ−1𝑏−1𝑏+𝑁−1𝑏+ℓ−1^ℓ𝑏ℓ𝑏.ℓ+𝑏𝑁−1𝑏+𝑁−1 (163) This proves (161). Now, by summing the ℙ{𝑆⁺_𝑏=ℓ,𝜎⁺_𝑏<+∞}, 𝑏≤ℓ≤𝑏+𝑁−1, given by (161), we obtain that ℙ𝜎⁺_𝑏<+∞=(−1)^𝑏𝑏𝑏𝑏+𝑁−1^{𝑏+𝑁−1}_ℓ=𝑏(−1)^ℓℓ𝑏𝑁−1ℓ−𝑏=𝑏𝑏+𝑁−1^𝑁−1_ℓ=0(−1)^ℓℓ.ℓ+𝑏𝑁−1 (164) Writing 1/(ℓ+𝑏)=∫¹₀𝑥^ℓ+𝑏−1d𝑥, we see that ^𝑁−1_ℓ=0(−1)^ℓℓ=ℓ+𝑏𝑁−1_{10𝑁−1ℓ=0}(−1)^ℓℓ𝑥𝑁−1_ℓ𝑥^𝑏−1d𝑥=¹₀(1−𝑥)^𝑁−1𝑥^𝑏−1d()!()!𝑥=𝑁−1𝑏−1()!.𝑏+𝑁−1 (165)
Hence ℙ{𝜎⁺_𝑏<+∞}=1. The proof of Theorem 22 is finished.

In the sequel, when considering 𝑆⁺_𝑏, we will omit the condition 𝜎⁺_𝑏<+∞.

Example 23. Let us have a look on the particular values 1, 2, 3, 4 of 𝑁.
(i) Case 𝑁=1. Evidently, in this case 𝑆⁺_𝑏=𝑏 and then ℙ𝑆⁺_𝑏=𝑏=1. (166)
This is the case of the ordinary random walk!
(ii) Case 𝑁=2. In this case the pseudorandom variables 𝜉_𝑛, 𝑛∈ℕ^∗, have two-valued upward jumps. Then the overshooting place must be either 𝑏 or 𝑏+1: 𝑆⁺_𝑏∈{𝑏,𝑏+1}. We have that ℙ𝑆⁺_𝑏=𝑏=𝑏+1,ℙ𝑆⁺_𝑏=𝑏+1=−𝑏. (167)
Of course, we immediately see that ℙ{𝑆⁺_𝑏=𝑏}+ℙ{𝑆⁺_𝑏=𝑏+1}=1.
(iii) Case 𝑁=3. In this case 𝑆⁺_𝑏∈{𝑏,𝑏+1,𝑏+2} and ℙ𝑆⁺_𝑏=1=𝑏2(),ℙ𝑆𝑏+1)(𝑏+2_+𝑏(),ℙ𝑆=𝑏+1=−𝑏𝑏+2_+𝑏=1=𝑏+22𝑏(𝑏+1). (168)
We can easily check that ℙ{𝑆⁺_𝑏=𝑏}+ℙ{𝑆⁺_𝑏=𝑏+1}+ℙ{𝑆⁺_𝑏=𝑏+2}=1.
(iv) Case 𝑁=4. In this case 𝑆⁺_𝑏∈{𝑏,𝑏+1,𝑏+2,𝑏+3} and ℙ𝑆⁺_𝑏=1=𝑏6(),ℙ𝑆𝑏+1)(𝑏+2)(𝑏+3_+𝑏1=𝑏+1=−2𝑏(),ℙ𝑆𝑏+2)(𝑏+3_+𝑏=1=𝑏+22𝑏(),ℙ𝑆𝑏+1)(𝑏+3_+𝑏1=𝑏+3=−6𝑏(𝑏+1)(𝑏+2). (169) We can easily check that ℙ{𝑆⁺_𝑏=𝑏}+ℙ{𝑆⁺_𝑏=𝑏+1}+ℙ{𝑆⁺_𝑏=𝑏+2}+ℙ{𝑆⁺_𝑏=𝑏+3}=1.

3.1.3. Pseudomoments of 𝑆⁺_𝑏

In the sequel, we use the notation (𝑖)_𝑛=𝑖(𝑖−1)(𝑖−2)⋯(𝑖−𝑛+1) for any 𝑖∈ℤ and any 𝑛∈ℕ^∗ and (𝑖)₀=1. Of course, (𝑖)_𝑛=𝑖!/(𝑖−𝑛)! and (𝑖)_𝑛/𝑛!=^𝑖_𝑛 if 𝑖≥𝑛. We also use the conventions 1/𝑖!=0 for any negative integer 𝑖 and ∑^𝑗_𝑘=𝑖=0 if 𝑖>𝑗.

In this section, we compute several functionals related to the pseudomoments of 𝑆⁺_𝑏. More precisely, we provide formulae for 𝔼[(𝑆⁺_𝑏−𝛽)_𝑛] (Theorem 25), 𝔼[(𝑆⁺_𝑏−𝑏)_𝑛] (Corollary 26), 𝔼[(𝑆⁺_𝑏)_𝑛], and 𝔼[(𝑆⁺_𝑏)^𝑛] (Theorem 27).

Putting the elementary identity 1/(ℓ+𝑏)=∫¹₀𝑥^ℓ+𝑏−1d𝑥 into the equality 𝔼𝑓𝑆⁺_𝑏=𝑏𝑏+𝑁−1^{𝑏+𝑁−1}_ℓ=𝑏(−1)^𝑏+ℓ𝑏ℓ𝑓(ℓ)𝑏𝑁−1ℓ−𝑏=𝑏𝑏+𝑁−1^𝑁−1_ℓ=0(−1)^ℓℓ𝑁−1𝑓(ℓ+𝑏),ℓ+𝑏 (170) we get the following integral representation of 𝔼[𝑓(𝑆⁺_𝑏)].

Theorem 24. For any function 𝑓 defined on {𝑏,…,𝑏+𝑁−1}, 𝔼𝑓𝑆⁺_𝑏𝑏×=𝑏𝑏+𝑁−1_{10𝑁−1ℓ=0}(−1)^ℓℓ𝑓()𝑥𝑁−1ℓ+𝑏^ℓ𝑥^𝑏−1d𝑥. (171)

Theorem 25. For any integers 𝑛≥0 and 𝛽, the factorial pseudo-moment of (𝑆⁺_𝑏−𝛽) of order 𝑛 is given by 𝔼𝑆⁺_𝑏−𝛽_𝑛=𝑏−𝛽!(𝑏−1)!^{𝑛∧(𝑁−1)}_𝑘=0∨∑_{𝑛+𝛽−𝑏}(−1)^𝑘(𝑘+𝑏−1)!!𝑛𝑘()𝑘+𝑏−𝛽−𝑛𝑖𝑓𝛽≤𝑏,−1^𝑛()!!×𝑏−1𝛽−𝑏−1_{𝑛∧(𝑁−1)𝑘}∑₌₀()!!𝑛𝑘𝑘+𝑏−1𝛽−𝑏+𝑛−𝑘−1𝑖𝑓𝛽≥𝑏+1. (172) If 𝑛≤𝑁−1, we simply have that 𝔼𝑆⁺_𝑏−𝛽_𝑛=−𝛽_𝑛=(−1)^𝑛𝛽𝛽+1⋯𝛽+𝑛−1. (173)

Proof. By (171), we have that 𝔼𝑆⁺_𝑏−𝛽_𝑛𝑏×=𝑏𝑏+𝑁−1_{10𝑁−1ℓ=0}(−1)^ℓℓ𝑁−1ℓ+𝑏−𝛽_𝑛𝑥^ℓ+𝑏−1d𝑥. (174) Next, by observing that (ℓ+𝑏−𝛽)_𝑛𝑥^ℓ+𝑏−1=𝑥^{𝑛+𝛽−1}×(d^𝑛/d𝑥^𝑛)(𝑥^{ℓ+𝑏−𝛽}), we obtain that ^𝑁−1_ℓ=0(−1)^ℓℓ𝑁−1ℓ+𝑏−𝛽_𝑛𝑥^ℓ+𝑏−1=^𝑁−1_ℓ=0(−1)^ℓℓ𝑁−1d^𝑛d𝑥^𝑛𝑥^{ℓ+𝑏−𝛽}𝑥^{𝑛+𝛽−1}=d^𝑛d𝑥^𝑛𝑁−1_ℓ=0(−1)^ℓℓ𝑥𝑁−1_{ℓ+𝑏−𝛽}𝑥^{𝑛+𝛽−1}=d^𝑛d𝑥^𝑛(1−𝑥)^𝑁−1𝑥^𝑏−𝛽𝑥^{𝑛+𝛽−1}. (175) Applying Leibniz rule to (175), we see that ^𝑁−1_ℓ=0(−1)^ℓℓ𝑁−1ℓ+𝑏−𝛽_𝑛𝑥^ℓ+𝑏−1=^𝑛_𝑘=0𝑛𝑘d^𝑘d𝑥^𝑘(1−𝑥)^𝑁−1d^𝑛−𝑘d𝑥^𝑛−𝑘𝑥^𝑏−𝛽𝑥^{𝑛+𝛽−1}=^{𝑛∧(𝑁−1)}_𝑘=0(−1)^𝑘𝑛𝑘(𝑁−1)_𝑘𝑏−𝛽_𝑛−𝑘×(1−𝑥)^{𝑁−1−𝑘}𝑥^{𝑘+𝑏−1}. (176) Therefore, ¹₀^𝑁−1_ℓ=0(−1)^ℓℓ𝑁−1ℓ+𝑏−𝛽_𝑛𝑥^ℓ+𝑏−1d𝑥=^{𝑛∧(𝑁−1)}_𝑘=0(−1)^𝑘(𝑁−1)_𝑘𝑏−𝛽_𝑛−𝑘𝑛𝑘×¹₀(1−𝑥)^{𝑁−1−𝑘}𝑥^{𝑘+𝑏−1}d𝑥=^{𝑛∧(𝑁−1)}_𝑘=0(−1)^𝑘(𝑁−1)_𝑘𝑏−𝛽_𝑛−𝑘𝑛𝑘×(𝑁−1−𝑘)!(𝑘+𝑏−1)!()!=!()!𝑏+𝑁−1𝑏−𝛽𝑁−1()!×𝑏+𝑁−1_{𝑛∧(𝑁−1)𝑘=0∨}∑_{𝑛+𝛽−𝑏}(−1)^𝑘(𝑘+𝑏−1)!!𝑛𝑘+𝑏−𝛽−𝑛𝑘if()𝛽≤𝑏,−1^𝑛(𝑁−1)!!()!×𝛽−𝑏−1𝑏+𝑁−1_{𝑛∧(𝑁−1)𝑘}∑₌₀()!!𝑛𝑘𝑘+𝑏−1𝛽−𝑏+𝑛−𝑘−1if𝛽≥𝑏+1. (177)
Finally, plugging (177) into (175) and (174) yields (172).
Assume now that 𝑛≤𝑁−1 and 𝛽≤𝑏. If 𝑛≥1−𝛽, we can write in (172) that (𝑘+𝑏−1)!!=𝑘+𝑏−𝛽−𝑛d^{𝛽+𝑛−1}d𝑥^{𝛽+𝑛−1}𝑥^{𝑘+𝑏−1}_𝑥=1. (178) Then, ^{𝑛∧(𝑁−1)}_{𝑘=0∨𝑛+𝛽−𝑏}(−1)^𝑘(𝑘+𝑏−1)!!𝑛𝑘=𝑘+𝑏−𝛽−𝑛_𝑛𝑘=0(−1)^𝑘(𝑘+𝑏−1)!!𝑛𝑘=𝑘+𝑏−𝛽−𝑛d^{𝛽+𝑛−1}d𝑥^{𝛽+𝑛−1𝑛}_𝑘=0(−1)^𝑘𝑛𝑘𝑥^𝑘𝑥^𝑏−1_𝑥=1=d^{𝛽+𝑛−1}d𝑥^{𝛽+𝑛−1}(1−𝑥)^𝑛𝑥^𝑏−1_𝑥=1=^𝑛_𝑘=0(−1)^𝑘(𝑛)_𝑘(𝑏−1)_{𝛽+𝑛−𝑘−1}𝑘×()𝛽+𝑛−11−𝑥^𝑛−𝑘𝑥^{𝑘+𝑏−𝛽−𝑛}_𝑥=1=(−1)^𝑛(𝑛)_𝑛(𝑏−1)_𝛽−1𝑛=()𝛽+𝑛−1−1^𝑛(𝑏−1)!𝛽+𝑛−1!!!=()𝑏−𝛽𝛽−1−1^𝑛(𝑏−1)!!𝛽⋯.𝑏−𝛽𝛽+1𝛽+𝑛−1 (179) Putting (179) into (172) yields (173). If 𝑛≤−𝛽 (which requires that 𝛽≤0), in (172), we write instead that (𝑘+𝑏−1)!!=1𝑘+𝑏−𝛽−𝑛−𝑛−𝛽!¹₀𝑥^{𝑘+𝑏−1}(1−𝑥)^{−𝑛−𝛽}d𝑥. (180) Then, ^{𝑛∧(𝑁−1)}_{𝑘=0∨𝑛+𝛽−𝑏}(−1)^𝑘(𝑘+𝑏−1)!!𝑛𝑘=𝑘+𝑏−𝛽−𝑛_𝑛𝑘=0(−1)^𝑘(𝑘+𝑏−1)!!𝑛𝑘=1𝑘+𝑏−𝛽−𝑛!×−𝑛−𝛽_10𝑛𝑘=0(−1)^𝑘𝑛𝑘𝑥^𝑘𝑥^𝑏−1(1−𝑥)^{−𝑛−𝛽}d𝑥=1−𝑛−𝛽!¹₀𝑥^𝑏−1(1−𝑥)^−𝛽d𝑥=(𝑏−1)!−𝛽!!!=()𝑏−𝛽−𝑛−𝛽−1^𝑛(𝑏−1)!!𝛽⋯.𝑏−𝛽𝛽+1𝛽+𝑛−1 (181)
Putting (181) into (172) yields (173) in this case too.
Assume finally that 𝑛≤𝑁−1 and 𝛽≥𝑏+1. We write in (172) that ()!!=!𝑘+𝑏−1𝛽−𝑏+𝑛−𝑘−1𝛽+𝑛−1¹₀𝑥^{𝑘+𝑏−1}(1−𝑥)^{𝛽−𝑏+𝑛−𝑘−1}d𝑥. (182) Then ^{𝑛∧(𝑁−1)}_𝑘=0()!!𝑛𝑘=!×𝑘+𝑏−1𝛽−𝑏+𝑛−𝑘−1𝛽+𝑛−1_10𝑛𝑘=0𝑛𝑘𝑥1−𝑥^𝑘𝑥^𝑏−1(1−𝑥)^{𝛽−𝑏+𝑛−1}d𝑥=𝛽+𝑛−1!¹₀𝑥^𝑏−1(1−𝑥)^{𝛽−𝑏−1}d𝑥=()!!!𝑏−1𝛽−𝑏−1𝛽+𝑛−1!.𝛽−1 (183)
Putting (183) into (172) yields (173).

By choosing 𝛽=𝑏 in Theorem 25, we derive that 𝔼𝑆⁺_𝑏−𝑏_𝑛=1(𝑏−1)!^{𝑛∧(𝑁−1)}_𝑘=𝑛(−1)^𝑘(𝑘+𝑏−1)!()!𝑛𝑘.𝑘−𝑛 (184) We immediately obtain the following particular result which will be used in Theorem 28.

Corollary 26. The factorial pseudomoments of (𝑆⁺_𝑏−𝑏) are given by 𝔼𝑆⁺_𝑏−𝑏_𝑛=(−𝑏)_𝑛if00≤𝑛≤𝑁−1,if𝑛≥𝑁. (185)

The above identity can be rewritten, if 0≤𝑛≤𝑁−1, as 𝔼𝑆⁺_𝑏𝑛=()−𝑏−1^𝑛.𝑛+𝑏−1𝑏−1 (186)

Moreover, since 𝑆⁺_𝑏∈{𝑏,𝑏+1,…,𝑏+𝑁−1}, it is clear that (𝑆⁺_𝑏−𝑏)(𝑆⁺_𝑏−𝑏−1)⋯(𝑆⁺_𝑏−𝑏−𝑁+1)=0 which immediately entails that (𝑆⁺_𝑏−𝑏)_𝑛=0 for any 𝑛≥𝑁; then 𝔼[(𝑆⁺_𝑏−𝑏)_𝑛]=0 for 𝑛≥𝑁 as stated in Corollary 26.

By choosing 𝛽=0 in Theorem 25, we plainly extract that 𝔼[(𝑆⁺_𝑏)_𝑛]=0 for 𝑛∈{1,…,𝑁−1}. Moreover, as previously mentioned, (𝑆⁺_𝑏)_𝑛=0 for any 𝑛≥𝑏+𝑁; then 𝔼[(𝑆⁺_𝑏)_𝑛]=0 for 𝑛≥𝑏+𝑁. Actually, we can compute the factorial pseudomoments of 𝑆⁺_𝑏, 𝔼[(𝑆⁺_𝑏)_𝑛], for 𝑛∈{𝑁,𝑁+1,…,𝑏+𝑁}. The formula of Theorem 25 seems to be untractable, so we provide another way for evaluating them.

Theorem 27. The factorial pseudomoments of 𝑆⁺_𝑏 are given by 𝔼𝑆⁺_𝑏𝑛=(−1)^𝑁−1()𝑛−1𝑁−1𝑏+𝑁−1_𝑛𝑖𝑓𝑁≤𝑛≤𝑏+𝑁−1,0𝑖𝑓1≤𝑛≤𝑁−1𝑜𝑟𝑛≥𝑏+𝑁. (187) Moreover, for 𝑛∈{1,…,𝑁−1}, the pseudo-moment of 𝑆⁺_𝑏 of order 𝑛 vanishes: 𝔼𝑆⁺_𝑏^𝑛𝔼𝑆=0,_+𝑏𝑁=(−1)^𝑁−1(𝑏+𝑁−1)!()!()𝑏−1=−−𝑏_𝑁. (188)

Proof. We focus on the case where 𝑁≤𝑛≤𝑏+𝑁−1. We have that 𝔼𝑆⁺_𝑏𝑛=^𝑁−1_ℓ=0ℙ𝑆⁺_𝑏=ℓ+𝑏(ℓ+𝑏)_𝑛𝑏=𝑏𝑏+𝑁−1^𝑁−1_ℓ=0(−1)^ℓ(ℓ+𝑏−1)_𝑛−1ℓ.𝑁−1 (189)
The intermediate sum lying in the last displayed equality can be evaluated as follows: by observing that (ℓ+𝑏−1)_𝑛−1=(d^𝑛−1/d𝑥^𝑛−1)(𝑥^ℓ+𝑏−1)|_𝑥=1 and appealing to Leibniz rule, we obtain that ^𝑁−1_ℓ=0(−1)^ℓ(ℓ+𝑏−1)_𝑛−1ℓ=𝑁−1d^𝑛−1d𝑥^{𝑛−1𝑁−1}_ℓ=0(−1)^ℓℓ𝑥𝑁−1_{ℓ+𝑏−1𝑥=1}=d^𝑛−1d𝑥^𝑛−1(1−𝑥)^𝑁−1𝑥^𝑏−1_𝑥=1=^𝑛−1_{𝑗=𝑁−1}𝑗𝑛−1d^𝑗d𝑥^𝑗(1−𝑥)^𝑁−1_𝑥=1d^{𝑛−1−𝑗}d𝑥^{𝑛−1−𝑗}𝑥^𝑏−1_𝑥=1=(−1)^𝑁−1(𝑁−1)!()𝑛−1𝑁−1𝑏−1_𝑛−𝑁=(−1)^𝑁−1(𝑏−1)!(𝑛−1)!()!()!.𝑛−𝑁𝑏+𝑁−𝑛−1 (190) Consequently, 𝔼𝑆⁺_𝑏𝑛=(−1)^𝑁−1𝑏𝑏𝑏+𝑁−1(𝑏−1)!(𝑛−1)!()!()!=()𝑛−𝑁𝑏+𝑁−𝑛−1−1^𝑁−1(𝑛−1)!(𝑏+𝑁−1)!()!()!()!.𝑛−𝑁𝑁−1𝑏+𝑁−𝑛−1 (191) This is the result announced in Theorem 27 when 𝑁≤𝑛≤𝑏+𝑁−1.
Next, concerning the pseudomoments of 𝑆⁺_𝑏, we appeal to an elementary argument of linear algebra: the family (1,𝑋₁,𝑋₂,…,𝑋_𝑛) (recall that 𝑋_𝑘=𝑋(𝑋−1)⋯(𝑋−𝑘+1)) is a basis of the space of polynomials of degree not greater than 𝑛. So, 𝑋^𝑛 can be written as a linear combination of 𝑋₁,𝑋₂,…,𝑋_𝑛. Then 𝔼[(𝑆⁺_𝑏)^𝑛] can be written as a linear combination of the factorial pseudomoments of 𝑆⁺_𝑏 of order between 1 and 𝑛. The latters cancel for 𝑛∈{1,…,𝑁−1}. As a result, 𝔼[(𝑆⁺_𝑏)^𝑛]=0.
The same argument ensures the equalities 𝔼[(𝑆⁺_𝑏)_𝑁]=𝔼[(𝑆⁺_𝑏)^𝑁], which is equal to (−1)^𝑁−1(𝑏+𝑁−1)_𝑁, and 𝔼[(𝑆⁺_𝑏−𝑏)_𝑁]=𝔼[(𝑆⁺_𝑏)^𝑁]+(−𝑏)^𝑁 which vanishes. Each of them yields the value of 𝔼[(𝑆⁺_𝑏)^𝑁].
The proof of Theorem 27 is completed.

3.2. Link with the High-Order Finite-Difference Operator

Set Δ⁺𝑓(𝑖)=𝑓(𝑖+1)−𝑓(𝑖) for any 𝑖∈ℤ and (Δ⁺)^𝑗=Δ⁺∘⋯∘Δ⁺_𝑗times for any 𝑗∈ℕ^∗. Set also (Δ⁺)⁰𝑓=𝑓. The quantity (Δ⁺)^𝑗 is the iterated forward finite-difference operator given by Δ^+𝑗𝑓(𝑖)=^𝑗_𝑘=0(−1)^𝑗+𝑘𝑗𝑘𝑓(𝑖+𝑘). (192) Conversely, 𝑓(𝑖+𝑘) can be expressed by means of (Δ⁺)^𝑗𝑓(𝑖), 0≤𝑗≤𝑘, according to 𝑓(𝑖+𝑘)=^𝑘_𝑗=0𝑘𝑗Δ^+𝑗𝑓(𝑖). (193) We have the following expression for any functional of the pseudorandom variable 𝑆⁺_𝑏.

Theorem 28. One has, for any function 𝑓 defined on {𝑏,𝑏+1,…,𝑏+𝑁−1}, that 𝔼𝑓𝑆⁺_𝑏=^𝑁−1_𝑗=0(−1)^𝑗Δ𝑗+𝑏−1𝑏−1^+𝑗𝑓(𝑏). (194)

Proof. By (193), we see that 𝔼𝑓𝑆⁺_𝑏=𝔼^{𝑆+𝑏−𝑏}_𝑗=0𝑆⁺_𝑏𝑗Δ−𝑏_+𝑗𝑓(𝑏)=^𝑁−1_𝑗=0𝔼𝑆⁺_𝑏𝑗Δ−𝑏_+𝑗𝑓(𝑏) (195) which immediately yields (194) thanks to (186).

Corollary 29. The generating function of 𝑆⁺_𝑏 is given by 𝔼𝜁^𝑆+𝑏=𝜁^𝑏𝑁−1_𝑗=0()𝑗+𝑏−1𝑏−11−𝜁^𝑗. (196)

Proof. Let us apply Theorem 28 to the function 𝑓(𝑖)=𝜁^𝑖 for which we plainly have (Δ⁺)^𝑗𝑓(𝑖)=(−1)^𝑗𝜁^𝑖(1−𝜁)^𝑗. This immediately yields (196).

Remark 30. A direct computation with (161) yields the alternative representation: 𝔼𝜁^𝑆+𝑏𝑏=𝑏𝜁𝑏+𝑁−1_𝑏10𝑥^𝑏−1(1−𝜁𝑥)^𝑁−1d𝑥. (197)
Of special interest is the case when the starting point of the pseudorandom walk is any point 𝑥∈ℤ. By translating 𝑏 into 𝑏−𝑥 and the function 𝑓 into the shifted function 𝑓(⋅+𝑥) in formula (194), we get that 𝔼_𝑥𝑓𝑆=⁺_𝑏^def𝔼𝑓𝑥+𝑆⁺_𝑏−𝑥=^𝑁−1_𝑗=0(−1)^𝑗𝑗Δ𝑗+𝑏−𝑥−1_+𝑗𝑓(𝑏). (198) Thus, we obtain the following result.

Theorem 31. One has, for any function 𝑓 defined on {𝑏,𝑏+1,…,𝑏+𝑁−1}, that 𝔼_𝑥𝑓𝑆⁺_𝑏=^𝑁−1_𝑗=0𝑃⁺_𝑏,𝑗(𝑥)Δ^+𝑗𝑓(𝑏) (199) with 𝑃⁺_𝑏,0(𝑥)=1 and, for 𝑗∈{1,…,𝑁−1}, 𝑃⁺_𝑏,𝑗(𝑥)=1𝑗!^𝑗−1_𝑘=0(𝑥−𝑏−𝑘). (200) The 𝑃⁺_𝑏,𝑗, 0≤𝑗≤𝑁−1, are Newton interpolating polynomials. They are of degree not greater than (𝑁−1) and characterized, for any 𝑘∈{0,…,𝑁−1}, by Δ^+𝑘𝑃⁺_𝑏,𝑗(𝑏)=𝛿_𝑗𝑘. (201)

Proof. Coming back to the proof of Theorem 28 and appealing to Theorem 22, we write that 𝑃⁺_𝑏,𝑗(𝑥)=𝔼_𝑥𝑆⁺_𝑏𝑗𝑆−𝑏=𝔼^+𝑏−𝑥𝑗=+𝑥−𝑏_{𝑁−1𝑚=𝑗}𝑚𝑗ℙ𝑆⁺_𝑏−𝑥=1=𝑏−𝑥+𝑚(𝑁−1)!^𝑁−1_𝑚=𝑗(−1)^𝑚𝑚𝑗𝑚𝐾𝑁−1_𝑚(𝑥), (202) where, for any 𝑚∈{0,…,𝑁−1}, 𝐾_𝑚(𝑥)=∏^𝑁−1_𝑘=0(𝑏−𝑥+𝑘)()=𝑏−𝑥+𝑚_{0≤𝑘≤𝑁−1𝑘≠𝑚}(𝑏−𝑥+𝑘). (203) The expression 𝐾_𝑚(𝑥) defines a polynomial of the variable 𝑥 of degree (𝑁−1), so 𝑃⁺_𝑏,𝑗 is a polynomial of degree not greater than (𝑁−1). It is obvious that 𝐾_𝑚(𝑏+ℓ)=0 for ℓ∈{0,…,𝑁−1}⧵{𝑚}. On the other hand, 𝐾_ℓ(𝑏+ℓ)=(−1)^ℓ(𝑁−1)!/_ℓ^𝑁−1. By putting this into (202), we get that 𝑃⁺_𝑏,𝑗()=ℓ𝑗𝑏+ℓ𝟙_{𝑗≤ℓ}. (204) Next, we obtain, for any 𝑘∈{0,…,𝑁−1}, that Δ^+𝑘𝑃⁺_𝑏,𝑗(𝑏)=^𝑘_ℓ=0(−1)^𝑘+ℓ𝑘ℓ𝑃⁺_𝑏,𝑗()=𝑏+ℓ_𝑘ℓ=𝑗(−1)^𝑘+ℓ𝑘ℓℓ𝑗=(−1)^𝑗+𝑘𝑘𝑗^{𝑘−𝑗ℓ=0}(−1)^ℓℓ𝑘−𝑗=𝛿_𝑗𝑘. (205)
The proof of Theorem 31 is finished.

We complete this paragraph by stating a strong pseudo-Markov property related to time 𝜎⁺_𝑏.

Theorem 32. One has, for any function 𝑓 defined on ℤ and any 𝑛∈ℕ, that 𝔼_𝑥𝑓𝑆_𝜎⁺_𝑏+𝑛=^𝑁−1_𝑗=0𝑃⁺_𝑏,𝑗(𝑥)Δ^+𝑗𝔼_𝑏𝑓𝑆_𝑛. (206) In (206), the operator (Δ⁺)^𝑗 acts on the variable 𝑏.

Proof. We denote by ℙ_𝑥 the pseudoprobability associated with the pseudoexpectation 𝔼_𝑥. Actually, it represents the pseudoprobability related to the pseudorandom walk started at point 𝑥 at time 0. We have, by independence of the 𝜉_𝑗’s, that 𝔼_𝑥𝑓𝑆_𝜎⁺_𝑏+𝑛=_{𝑘,ℓ∈ℕ∶𝑏≤ℓ≤𝑏+𝑁−1}𝔼_𝑥𝟙_{𝜎⁺_{𝑏=𝑘,𝑆}⁺_𝑏=ℓ}𝑓𝑆⁺_𝑏+𝜉_𝑘+1+⋯+𝜉_𝑘+𝑛=_{𝑘,ℓ∈ℕ∶𝑏≤ℓ≤𝑏+𝑁−1}ℙ_𝑥𝜎⁺_𝑏=𝑘,𝑆⁺_𝑏=ℓ𝔼𝑓ℓ+𝜉₁+⋯+𝜉_𝑛=^{𝑏+𝑁−1}_ℓ=𝑏ℙ_𝑥𝑆⁺_𝑏=ℓ𝔼_ℓ𝑓𝑆_𝑛=𝔼_𝑥𝔼_𝑆⁺_𝑏𝑓𝑆_𝑛. (207) Hence, by setting 𝑔(𝑥)=𝔼_𝑥[𝑓(𝑆_𝑛)], we have obtained that 𝔼_𝑥𝑓𝑆_𝜎⁺_𝑏+𝑛=𝔼_𝑥𝑔𝑆⁺_𝑏 (208) which proves (206) thanks to (199).

Example 33. Below, we display the form of (206) for the particular values 1, 2, 3 of 𝑁. (i)For 𝑁=1, (206) reads 𝔼_𝑥𝑓𝑆_𝜎⁺_𝑏+𝑛=𝔼_𝑏𝑓𝑆_𝑛 (209) which is of course trivial! This is the strong Markov property for the ordinary random walk.(ii)For 𝑁=2, (206) reads 𝔼_𝑥𝑓𝑆_𝜎⁺_𝑏+𝑛=𝔼_𝑏𝑓𝑆_𝑛+(𝑥−𝑏)Δ⁺𝔼_𝑏𝑓𝑆_𝑛=(𝑏−𝑥+1)𝔼_𝑏𝑓𝑆_𝑛+(𝑥−𝑏)𝔼_𝑏+1𝑓𝑆_𝑛. (210)(iii)For 𝑁=3, (206) reads 𝔼_𝑥𝑓𝑆_𝜎⁺_𝑏+𝑛=𝔼_𝑏𝑓𝑆_𝑛+(𝑥−𝑏)Δ⁺𝔼_𝑏𝑓𝑆_𝑛+12(𝑥−𝑏)(𝑥−𝑏−1)Δ⁺²𝔼_𝑏𝑓𝑆_𝑛=12(𝑥−𝑏−1)(𝑥−𝑏−2)𝔼_𝑏𝑓𝑆_𝑛−(𝑥−𝑏)(𝑥−𝑏−2)𝔼_𝑏+1𝑓𝑆_𝑛+12(𝑥−𝑏)(𝑥−𝑏−1)𝔼_𝑏+2𝑓𝑆_𝑛. (211)

3.3. Joint Pseudodistribution of (𝜏⁺_𝑏,𝑋⁺_𝑏)

Below, we give an ad hoc definition for the convergence of a family of exit times.

Definition 34. Let ((𝑋^𝜀_𝑡)_𝑡≥0)_𝜀>0 be a family of pseudoprocesses which converges towards a pseudoprocess (𝑋_𝑡)_𝑡≥0 when 𝜀→0⁺ in the sense of Definition 13. Let 𝐼 be a subset of ℝ and set 𝜏^𝜀_𝐼=inf{𝑡≥0∶𝑋^𝜀_𝑡∉𝐼}, 𝑋^𝜀_𝐼=𝑋^𝜀_𝜏^𝜀_𝐼 and 𝜏_𝐼=inf{𝑡≥0∶𝑋_𝑡∉𝐼}, 𝑋_𝐼=𝑋_𝜏𝐼.
We say that 𝜏^𝜀_𝐼,𝑋^𝜀_𝐼⟶_𝜀→0+𝜏_𝐼,𝑋_𝐼 (212) if and only if 𝔼∀𝜆>0,∀𝜇∈ℝ,e^{−𝜆𝜏𝜀𝐼+𝑖𝜇𝑋𝜀𝐼}𝟙_{𝜏^𝜀_𝐼<+∞}⟶_𝜀→0+𝔼e^{−𝜆𝜏𝐼+𝑖𝜇𝑋𝐼}𝟙_{{𝜏𝐼<+∞}}. (213) We say that 𝑋^𝜀_𝐼⟶_𝜀→0+𝑋_𝐼 (214) if and only if ∀𝜇∈ℝ,𝔼e^{𝑖𝜇𝑋𝜀𝐼}𝟙_{𝜏^𝜀_𝐼<+∞}⟶_𝜀→0+𝔼e^{𝑖𝜇𝑋𝐼}𝟙_{{𝜏𝐼<+∞}}. (215)
As in Section 2.3, we choose for the family ((𝑋^𝜀_𝑡)_𝑡≥0)_𝜀>0 the pseudoprocesses defined, for any 𝜀>0, by 𝑋^𝜀_𝑡=𝜀𝑆_{⌊𝑡/𝜀2𝑁⌋},𝑡≥0, (216) and for the pseudoprocess (𝑋_𝑡)_𝑡≥0 the pseudo-Brownian motion. For 𝐼, we choose the interval (−∞,𝑏) so that 𝜏^𝜀_𝐼=𝜏_𝑏^𝜀+, 𝑋^𝜀_𝐼=𝑋_𝑏^𝜀+ and 𝜏_𝐼=𝜏⁺_𝑏, 𝑋_𝐼=𝑋⁺_𝑏. Set 𝑏_𝜀=⌈𝑏/𝜀⌉ where ⌈⋅⌉ is the usual ceiling function. We have 𝜏_𝑏^𝜀+=𝜀^2𝑁𝜎⁺_𝑏𝜀 and 𝑋_𝑏^𝜀+=𝜀𝑆⁺_𝑏𝜀. Recall the setting 𝜑_𝑗=−𝑖e^{𝑖((2𝑗−1)/2𝑁)𝜋}, 1≤𝑗≤𝑁.

Theorem 35. Assume that 𝑐≤1/2^2𝑁−1. The following convergence holds: 𝜏_𝑏^𝜀+,𝑋_𝑏^𝜀+⟶_𝜀→0+𝜏⁺_𝑏,𝑋⁺_𝑏, (217) where, for any 𝜆>0 and any 𝜇∈ℝ, 𝔼e^{−𝜆𝜏+𝑏+𝑖𝜇𝑋+𝑏}𝟙_{{𝜏𝑏<+∞}}=e^{𝑁𝑖𝜇𝑏}_{𝑘=11≤𝑗≤𝑁𝑗≠𝑘}𝜑_𝑗𝜑_𝑗−𝜑_{𝑘1≤𝑗≤𝑁𝑗≠𝑘}𝑖1−𝜑_𝑗𝜇_2𝑁𝜆/𝑐e^{−𝜑𝑘2𝑁𝜆/𝑐𝑏}. (218)

Proof. We already pointed out that the assumption 𝑐≤1/2^2𝑁−1 entails that 𝑀_∞=1. Therefore, (147) holds for 𝑧=e^{−𝜆𝜀2𝑁}<1/𝑀_∞=1, that is, for 𝜆>0. So, by (147), we have, for 𝜆>0, that 𝔼e^{−𝜆𝜏𝑏𝜀++𝑖𝜇𝑋𝑏𝜀+}𝟙_{𝜏𝑏^𝜀+_<+∞}=𝔼e^{−𝜆𝜀2𝑁𝜎+𝑏𝜀+𝑖𝜇𝜀𝑆+𝑏𝜀}𝟙_{𝜎⁺_{𝑏𝜀<+∞}}=e^{𝑖𝜇𝜀𝑏𝜀𝑁}_𝑘=1𝐿_𝑘e^{−𝜆𝜀2𝑁},e^𝑖𝜇𝜀𝑢_𝑘e^{−𝜆𝜀2𝑁𝑏𝜀}. (219) Recall that we previously set 𝑢_𝑗(𝜆,𝜀)=𝑢_𝑗(e^{−𝜆𝜀2𝑁}). Thanks to asymptotics (119), we get that 𝑢_𝑘(𝜆,𝜀)−𝑢_𝑗(𝜆,𝜀)∼_𝜀→0+𝜑_𝑗−𝜑_𝑘^2𝑁𝜆𝑐𝜀,1−𝑢_𝑗(𝜆,𝜀)e^𝑖𝜇𝜀∼_𝜀→0+𝜑_𝑗^2𝑁𝜆𝑐−𝑖𝜇𝜀. (220) Thus, lim_𝜀→0+𝐿_𝑘e^{−𝜆𝜀2𝑁},e^𝑖𝜇𝜀=_{1≤𝑗≤𝑁𝑗≠𝑘}𝜑_𝑗^2𝑁𝜆/𝑐−𝑖𝜇𝜑_𝑗−𝜑_𝑘^2𝑁𝜆/𝑐=_{1≤𝑗≤𝑁𝑗≠𝑘}𝜑_𝑗𝜑_𝑗−𝜑_{𝑘1≤𝑗≤𝑁𝑗≠𝑘}𝑖1−𝜑_𝑗𝜇_2𝑁.𝜆/𝑐 (221) Finally, we can easily conclude with the help of the elementary limits that lim_𝜀→0+e^{𝑖𝜇𝜀𝑏𝜀}=e^𝑖𝜇𝑏,lim_𝜀→0+𝑢_𝑘(𝜆,𝜀)^𝑏𝜀=e^{−𝜑𝑘2𝑁𝜆/𝑐𝑏}. (222)

Theorem 36. The following convergence holds: 𝑋_𝑏^𝜀+⟶_𝜀→0+𝑋⁺_𝑏, (223) where, for any 𝜇∈ℝ, 𝔼e^{𝑖𝜇𝑋+𝑏}𝟙_{𝜏⁺_𝑏<+∞}=e^{𝑖𝜇𝑏𝑁−1}_𝑗=0−𝑖𝜇𝑏^𝑗.𝑗! (224) This is the Fourier transform of the pseudorandom variable 𝑋⁺_𝑏. Moreover, ℙ𝜏⁺_𝑏<+∞=1. (225)

Proof. By (194), we have that 𝔼e^{𝑖𝜇𝑋𝑏𝜀+}𝟙_{𝜏𝑏^𝜀+_<+∞}=𝔼e^{𝑖𝜇𝜀𝑆+𝑏𝜀}𝟙_{𝜎⁺_{𝑏𝜀<+∞}}=e^{𝑖𝜇𝜀𝑏𝜀𝑁−1}_𝑘=0𝑗+𝑏_𝜀𝑏−1_𝜀−11−e^{𝑖𝜇𝜀𝑗}. (226) We can easily conclude by using the elementary asymptotics that lim_𝜀→0+e^{𝑖𝜇𝜀𝑏𝜀}=e^𝑖𝜇𝑏,𝑗+𝑏_𝜀𝑏−1_𝜀∼−1^𝜀→0+𝑏^𝑗𝑗!𝜀^𝑗,1−e^{𝑖𝜇𝜀𝑗}∼_𝜀→0+−𝑖𝜇𝜀^𝑗. (227)

Corollary 37. The pseudodistribution of 𝑋⁺_𝑏 is given by ℙ𝑋⁺_𝑏∈d𝑧d𝑧=^𝑁−1_𝑗=0𝑏^𝑗𝛿𝑗!^𝑏(𝑗)(𝑧). (228)

This formula should be understood as follows: for any (𝑁−1)-times differentiable function 𝑓, by omitting the condition 𝜏⁺_𝑏<+∞, 𝔼𝑓𝑋⁺_𝑏=^𝑁−1_𝑗=0(−1)^𝑗𝑏^𝑗𝑓𝑗!^(𝑗)(𝑏). (229) We retrieve a result of [11] and, in the case 𝑁=2, a pioneering result of [18].

4. Part III—First Exit Time from a Bounded Interval

4.1. On the Pseudodistribution of (𝜎_𝑎𝑏,𝑆_𝑎𝑏)

Let 𝑎, 𝑏 be two integers such that 𝑎<0<𝑏 and let ℰ={𝑎,𝑎−1,…,𝑎−𝑁+1}∪{𝑏,𝑏+1,…,𝑏+𝑁−1}. In this section, we explicitly compute the generating function of (𝜎_𝑎𝑏,𝑆_𝑎𝑏). Set, for ℓ∈ℰ, 𝐻_𝑎𝑏,ℓ(𝑧)=𝔼𝑧^𝜎𝑎𝑏𝟙_{{𝑆𝑎𝑏=ℓ,𝜎𝑎𝑏<+∞}}=_𝑘∈ℕℙ𝜎_𝑎𝑏=𝑘,𝑆_𝑎𝑏=ℓ𝑧^𝑘. (230)

We are able to provide an explicit expression of 𝐻_𝑎𝑏,ℓ(𝑧). As in Section 3.1, due to (62), we have the following a priori estimate: |ℙ{𝜎_𝑎𝑏=𝑘,𝑆_𝑎𝑏=ℓ}|≤|ℙ{𝑆₁∈(𝑎,𝑏),…,𝑆_𝑘−1∈(𝑎,𝑏),𝑆_𝑘=ℓ}|≤𝑀^𝑘₁. As a byproduct, the power series defining 𝐻_𝑎𝑏,ℓ(𝑧) absolutely converges for |𝑧|<1/𝑀₁.

4.1.1. Joint Pseudodistribution of (𝜎_𝑎𝑏,𝑆_𝑎𝑏)

Theorem 38. The pseudodistribution of (𝜎_𝑎𝑏,𝑆_𝑎𝑏) is characterized by the identity, valid for any 𝑧∈(0,1), 𝔼𝑧^𝜎𝑎𝑏𝟙_{{𝑆𝑎𝑏=ℓ,𝜎𝑎𝑏<+∞}}=𝑈_ℓ𝑢₁(𝑧),…,𝑢_2𝑁(𝑧)𝑈𝑢₁(𝑧),…,𝑢_2𝑁(𝑧), (231) where 𝑈𝑢₁,…,𝑢_2𝑁=1𝑢₁⋯𝑢₁^𝑁−1𝑢₁^{𝑏−𝑎+𝑁−1}⋯𝑢₁^{𝑏−𝑎+2𝑁−2}⋮⋮⋮⋮⋮1𝑢_2𝑁⋯𝑢^𝑁−1_2𝑁𝑢^{𝑏−𝑎+𝑁−1}_2𝑁⋯𝑢^{𝑏−𝑎+2𝑁−2}_2𝑁 (232) and, if 𝑎−𝑁+1≤ℓ≤𝑎, 𝑈_ℓ(𝑢₁,…,𝑢_2𝑁) is the determinant1𝑢₁⋯𝑢₁^{ℓ+𝑁−𝑎−2}𝑢₁^{𝑁−𝑎−1}𝑢₁^{ℓ+𝑁−𝑎}⋯𝑢₁^𝑁−1𝑢₁^{𝑏−𝑎+𝑁−1}⋯𝑢₁^{𝑏−𝑎+2𝑁−2}⋮⋮⋮⋮⋮⋮⋮⋮1𝑢_2𝑁⋯𝑢^{ℓ+𝑁−𝑎−2}_2𝑁𝑢^{𝑁−𝑎−1}_2𝑁𝑢^{ℓ+𝑁−𝑎}_2𝑁⋯𝑢^𝑁−1_2𝑁𝑢^{𝑏−𝑎+𝑁−1}_2𝑁⋯𝑢^{𝑏−𝑎+2𝑁−2}_2𝑁, (233)and if 𝑏≤ℓ≤𝑏+𝑁−1, 𝑈_ℓ(𝑢₁,…,𝑢_2𝑁) is the determinant1𝑢₁⋯𝑢₁^𝑁−1𝑢₁^{𝑏−𝑎+𝑁−1}⋯𝑢₁^{ℓ+𝑁−𝑎−2}𝑢₁^{𝑁−𝑎−1}𝑢₁^{ℓ+𝑁−𝑎}⋯𝑢₁^{𝑏−𝑎+2𝑁−2}⋮⋮⋮⋮⋮⋮⋮⋮1𝑢_2𝑁⋯𝑢^𝑁−1_2𝑁𝑢^{𝑏−𝑎+𝑁−1}_2𝑁⋯𝑢^{ℓ+𝑁−𝑎−2}_2𝑁𝑢^{𝑁−𝑎−1}_2𝑁𝑢^{ℓ+𝑁−𝑎}_2𝑁⋯𝑢^{𝑏−𝑎+2𝑁−2}_2𝑁. (234)

Proof. Pick an integer 𝑘 such that 𝑘≤𝑎 or 𝑘≥𝑏. If 𝑆_𝑛=𝑘, then an exit of the interval (𝑎,𝑏) occurs before time 𝑛: 𝜎_𝑎𝑏≤𝑛. This remark and the independence of the increments of the pseudorandom walk entail that ℙ𝑆_𝑛=𝑘=ℙ𝑆_𝑛=𝑘,𝜎_𝑎𝑏=≤𝑛_{𝑛𝑗=0ℓ∈ℰ}ℙ𝑆_𝑛=𝑘,𝜎_𝑎𝑏=𝑗,𝑆_𝑎𝑏==ℓ_{𝑛𝑗=0ℓ∈ℰ}ℙ𝜎_𝑎𝑏=𝑗,𝑆_𝑎𝑏=ℓℙ𝑆_𝑛−𝑗=𝑘−ℓ. (235) Thanks to the absolute convergence of the series defining 𝐺_𝑘(𝑧) and 𝐻_𝑎𝑏,ℓ(𝑧) for 𝑧∈(0,1) and |𝑧|<1/𝑀₁, respectively, we can apply the generating function to equality (235). We get, for 𝑧∈(0,1/𝑀₁), that 𝐺_𝑘(𝑧)=_ℓ∈ℰ𝐺_𝑘−ℓ(𝑧)𝐻_𝑎𝑏,ℓ(𝑧). (236) Using expression (94) of 𝐺_𝑘, namely, 𝐺_𝑘(𝑧)=∑^𝑁_𝑗=1𝛼_𝑗(𝑧)𝑢_𝑗(𝑧)^|𝑘|, we get, for 𝑘≥𝑏+𝑁−1 (recall that 𝑣_𝑗(𝑧)=1/𝑢_𝑗(𝑧)), that ^𝑁_𝑗=1𝛼_𝑗(𝑧)𝑢_𝑗(𝑧)^𝑘_ℓ∈ℰ𝐻_𝑎𝑏,ℓ(𝑧)𝑣_𝑗(𝑧)^ℓ−1=0, (237) and, for 𝑘≤𝑎−𝑁+1, that ^𝑁_𝑗=1𝛼_𝑗(𝑧)𝑣_𝑗(𝑧)^𝑘_ℓ∈ℰ𝐻_𝑎𝑏,ℓ(𝑧)𝑢_𝑗(𝑧)^ℓ−1=0. (238) When limiting the range of 𝑘 to the set {𝑏+𝑁,𝑏+𝑁+1,…,𝑏+2𝑁−1} in (237) and to the set {𝑎−2𝑁+1,𝑎−2𝑁+2,…,𝑎−𝑁} in (238), we see that (237) and (238) are homogeneous Vandermonde systems whose solutions are trivial; that is, the terms within parentheses in (237) and (238) vanish. Thus, we get the two systems below: _ℓ∈ℰ𝐻_𝑎𝑏,ℓ(𝑧)𝑢_𝑗(𝑧)^ℓ=1,1≤𝑗≤𝑁,_ℓ∈ℰ𝐻_𝑎𝑏,ℓ(𝑧)𝑣_𝑗(𝑧)^ℓ=1,1≤𝑗≤𝑁. (239) It will be convenient to relabel the 𝑢_𝑗(𝑧)’s and 𝑣_𝑗(𝑧)’s, 1≤𝑗≤𝑁, as 𝑢_𝑗(𝑧)=𝑣_𝑗+𝑁(𝑧) and 𝑣_𝑗(𝑧)=𝑢_𝑗+𝑁(𝑧); note that 𝑣_𝑗(𝑧)=1/𝑢_𝑗(𝑧) for any 𝑗∈{1,…,2𝑁} and {𝑢₁(𝑧),…,𝑢_2𝑁(𝑧)}={𝑣₁(𝑧),…,𝑣_2𝑁(𝑧)}. By using the relabeling 𝑢_𝑗(𝑧),𝑣_𝑗(𝑧), 1≤𝑗≤2𝑁, we obtain the two equivalent following systems of 2𝑁 equations and 2𝑁 unknowns, 𝑢₁(𝑧),…,𝑢_2𝑁(𝑧) for the first one, 𝑣₁(𝑧),…,𝑣_2𝑁(𝑧) for the second one: _ℓ∈ℰ𝐻_𝑎𝑏,ℓ(𝑧)𝑢_𝑗(𝑧)^ℓ=1,1≤𝑗≤2𝑁,_ℓ∈ℰ𝐻_𝑎𝑏,ℓ(𝑧)𝑣_𝑗(𝑧)^ℓ=1,1≤𝑗≤2𝑁. (240)(241) Systems (240) and (241) are “lacunary” Vandermonde systems (some powers of 𝑢_𝑗(𝑧) are missing). For instance, let us rewrite system (240) as _ℓ∈ℰ𝐻_𝑎𝑏,ℓ(𝑧)𝑢_𝑗(𝑧)^{ℓ+𝑁−𝑎−1}=𝑢_𝑗(𝑧)^{𝑁−𝑎−1},1≤𝑗≤2𝑁. (242)
Cramer’s formulae immediately yield (231) at least for 𝑧∈(0,1/𝑀₁). By analyticity of the 𝑢_𝑗’s on (0,1), it is easily seen that (231) holds true for 𝑧∈(0,1). Systems (240) and (241) will be used in Lemma 42.

A method for computing the determinants exhibited in Theorem 38 and solving system (242) is proposed in Appendix A.1. In particular, we can deduce from Proposition A.3 an alternative representation of 𝔼(𝑧^𝜎𝑎𝑏𝟙_{{𝑆𝑎𝑏=ℓ,𝜎𝑎𝑏<+∞}}) which can be seen as the analogous of (124). Set 𝑠₀(𝑧)=1 and, for 𝑘,ℓ∈{1,…,2𝑁}, 𝑠_ℓ(𝑧)=_{1≤𝑖1<⋯<𝑖ℓ≤2𝑁}𝑢_𝑖1(𝑧)⋯𝑢_𝑖ℓ(𝑧),𝑝_𝑘(𝑧)=_{1≤𝑖≤2𝑁𝑖≠𝑘}𝑢_𝑘(𝑧)−𝑢_𝑖(𝑧), (243)𝑠_𝑘,0(𝑧)=1, 𝑠_𝑘,𝑚(𝑧)=0 for any integer 𝑚 such that 𝑚≤−1 or 𝑚≥2𝑁 and, for 𝑚∈{1,…,2𝑁−1}, 𝑠_𝑘,𝑚(𝑧)=_{1≤𝑖1<⋯<𝑖𝑚𝑖≤2𝑁1,…,𝑖𝑚≠𝑘}𝑢_𝑖1(𝑧)⋯𝑢_𝑖𝑚(𝑧). (244) Set also 𝑈(𝑧)=𝑠_𝑁(𝑧)𝑠_𝑁−1(𝑧)⋯𝑠_{𝑁−𝑏+𝑎+2}(𝑧)𝑠_𝑁+1(𝑧)𝑠_𝑁(𝑧)⋯𝑠_{𝑁−𝑏+𝑎+1}(𝑧)𝑠⋮⋮⋮_{𝑁+𝑏−𝑎−2}(𝑧)𝑠_{𝑁+𝑏−𝑎−3}(𝑧)⋯𝑠_𝑁(𝑧),𝑈_𝑘ℓ(𝑧)=𝑠_𝑘,𝑁(𝑧)𝑠_{𝑘,𝑁−1}(𝑧)⋯𝑠_{𝑘,𝑁−𝑏+𝑎+1}(𝑧)𝑠_𝑘,𝑁+1(𝑧)𝑠_𝑘,𝑁(𝑧)⋯𝑠_{𝑘,𝑁−𝑏+𝑎}(𝑧)𝑠⋮⋮⋮_{𝑘,𝑁+𝑏−𝑎−2}(𝑧)𝑠_{𝑘,𝑁+𝑏−𝑎−3}(𝑧)⋯𝑠_{𝑘,𝑁−1}(𝑧)𝑠_{𝑘,𝑁+𝑏−ℓ−1}(𝑧)𝑠_{𝑘,𝑁+𝑏−ℓ−2}(𝑧)⋯𝑠_{𝑘,𝑁+𝑎−ℓ}(𝑧). (245) Then, applying Proposition A.3 with the choices 𝑝=𝑟=𝑁 and 𝑞=𝑏−𝑎−1 leads, for any ℓ∈ℰ, to 𝔼𝑧^𝜎𝑎𝑏𝟙_{{𝑆𝑎𝑏=ℓ,𝜎𝑎𝑏<+∞}}=(−1)^{ℓ+𝑁−𝑎−1}𝑈(𝑧)^2𝑁_𝑘=1𝑈_𝑘ℓ(𝑧)𝑝_𝑘(𝑧)𝑢_𝑘(𝑧)^{𝑁−𝑎−1}. (246)

The double generating function defined by 𝔼𝑧^𝜎𝑎𝑏𝜁^𝑆𝑎𝑏𝟙_{{𝜎𝑎𝑏<+∞}}=_ℓ∈ℰ𝔼𝑧^𝜎𝑎𝑏𝟙_{{𝑆𝑎𝑏=ℓ,𝜎𝑎𝑏<+∞}}𝜁^ℓ (247) admits an interesting representation by means of interpolating polynomials that we display in the following theorem.

Theorem 39. The double generating function of (𝜎_𝑎𝑏,𝑆_𝑎𝑏) is given, for 𝑧∈(0,1/𝑀₁) and 𝜁∈ℂ, by 𝔼𝑧^𝜎𝑎𝑏𝜁^𝑆𝑎𝑏𝟙_{{𝜎𝑎𝑏<+∞}}=^2𝑁_𝑘=1𝐿_𝑘(𝑧,𝜁)𝑣_𝑘(𝑧)𝜁^{𝑎−𝑁+1}, (248) where, for 𝑘∈{1,…,2𝑁}𝐿_𝑘()=𝑈𝑢𝑧,𝜁¹(𝑧),…,𝑢_𝑘−1(𝑧),𝜁,𝑢_𝑘+1(𝑧),…,𝑢_2𝑁(𝑧)𝑈𝑢₁(𝑧),…,𝑢_2𝑁(𝑧). (249) The functions 𝜁↦𝐿_𝑘(𝑧,𝜁) are interpolating polynomials satisfying 𝐿_𝑘(𝑧,𝑢_𝑗(𝑧))=𝛿_𝑗𝑘 and can be expressed as 𝐿_𝑘(𝑧,𝜁)=𝑃_𝑘(𝑧,𝜁)_{1≤𝑗≤2𝑁𝑗≠𝑘}𝜁−𝑢_𝑗(𝑧)𝑢_𝑘(𝑧)−𝑢_𝑗(𝑧), (250) where 𝜁↦𝑃_𝑘(𝑧,𝜁) are some polynomials of degree (𝑏−𝑎−1).

Proof. By (231), we have that 𝔼𝑧^𝜎𝑎𝑏𝜁^𝑆𝑎𝑏𝟙_{{𝑆𝑎𝑏≤𝑎,𝜎𝑎𝑏<+∞}}=^𝑎_{ℓ=𝑎−𝑁+1}𝐻_𝑎𝑏,ℓ(𝑧)𝜁^ℓ=^𝑎_{ℓ=𝑎−𝑁+1}𝑈_ℓ𝑢₁(𝑧),…,𝑢_2𝑁(𝑧)𝑈𝑢₁(𝑧),…,𝑢_2𝑁(𝑧)𝜁^ℓ. (251) In order to simplify the text, we omit the variable 𝑧. We expand the determinant 𝑈_ℓ(𝑢₁,…,𝑢_2𝑁), 𝑎−𝑁+1≤ℓ≤𝑎 with respect to its (ℓ+𝑁−𝑎)th column: 𝑈_ℓ𝑢₁,…,𝑢_2𝑁=^2𝑁_𝑘=1𝑢_𝑘^{𝑁−𝑎−1}𝑈_𝑘ℓ𝑢₁,…,𝑢_𝑘−1,𝑢_𝑘+1,…,𝑢_2𝑁, (252)
where 𝑈_𝑘ℓ(𝑢₁,…,𝑢_𝑘−1,𝑢_𝑘+1,…,𝑢_2𝑁), 1≤𝑘≤2𝑁, is the determinant1𝑢₁⋯𝑢₁^{ℓ+𝑁−𝑎−2}0𝑢₁^{ℓ+𝑁−𝑎}⋯𝑢₁^𝑁−1𝑢₁^{𝑏−𝑎+𝑁−1}⋯𝑢₁^{𝑏−𝑎+2𝑁−2}⋮⋮⋮⋮⋮⋮⋮⋮1𝑢_𝑘−1⋯𝑢^{ℓ+𝑁−𝑎−2}_𝑘−10𝑢^{ℓ+𝑁−𝑎}_𝑘−1⋯𝑢^𝑁−1_𝑘−1𝑢^{𝑏−𝑎+𝑁−1}_𝑘−1⋯𝑢^{𝑏−𝑎+2𝑁−2}_𝑘−11𝑢_𝑘⋯𝑢_𝑘^{ℓ+𝑁−𝑎−2}1𝑢_𝑘^{ℓ+𝑁−𝑎}⋯𝑢_𝑘^𝑁−1𝑢_𝑘^{𝑏−𝑎+𝑁−1}⋯𝑢_𝑘^{𝑏−𝑎+2𝑁−2}1𝑢_𝑘+1⋯𝑢^{ℓ+𝑁−𝑎−2}_𝑘+10𝑢^{ℓ+𝑁−𝑎}_𝑘+1⋯𝑢^𝑁−1_𝑘+1𝑢^{𝑏−𝑎+𝑁−1}_𝑘+1⋯𝑢^{𝑏−𝑎+2𝑁−2}_𝑘+1⋮⋮⋮⋮⋮⋮⋮⋮1𝑢_2𝑁⋯𝑢^{ℓ+𝑁−𝑎−2}_2𝑁0𝑢^{ℓ+𝑁−𝑎}_2𝑁⋯𝑢^𝑁−1_2𝑁𝑢^{𝑏−𝑎+𝑁−1}_2𝑁⋯𝑢^{𝑏−𝑎+2𝑁−2}_2𝑁 (253)
which plainly coincides with1𝑢₁⋯𝑢₁^{ℓ+𝑁−𝑎−2}𝑢₁^{ℓ+𝑁−𝑎−1}𝑢₁^{ℓ+𝑁−𝑎}⋯𝑢₁^𝑁−1𝑢₁^{𝑏−𝑎+𝑁−1}⋯𝑢₁^{𝑏−𝑎+2𝑁−2}⋮⋮⋮⋮⋮⋮⋮⋮1𝑢_𝑘−1⋯𝑢^{ℓ+𝑁−𝑎−2}_𝑘−1𝑢^{ℓ+𝑁−𝑎−1}_𝑘−1𝑢^{ℓ+𝑁−𝑎}_𝑘−1⋯𝑢^𝑁−1_𝑘−1𝑢^{𝑏−𝑎+𝑁−1}_𝑘−1⋯𝑢^{𝑏−𝑎+2𝑁−2}_𝑘−100⋯010⋯00⋯01𝑢_𝑘+1⋯𝑢^{ℓ+𝑁−𝑎−2}_𝑘+1𝑢^{ℓ+𝑁−𝑎−1}_𝑘+1𝑢^{ℓ+𝑁−𝑎}_𝑘+1⋯𝑢^𝑁−1_𝑘+1𝑢^{𝑏−𝑎+𝑁−1}_𝑘+1⋯𝑢^{𝑏−𝑎+2𝑁−2}_𝑘+1⋮⋮⋮⋮⋮⋮⋮⋮1𝑢_2𝑁⋯𝑢^{ℓ+𝑁−𝑎−2}_2𝑁𝑢^{ℓ+𝑁−𝑎−1}_2𝑁𝑢^{ℓ+𝑁−𝑎}_2𝑁⋯𝑢^𝑁−1_2𝑁𝑢^{𝑏−𝑎+𝑁−1}_2𝑁⋯𝑢^{𝑏−𝑎+2𝑁−2}_2𝑁. (254)
Therefore, we obtain that ^𝑎_{ℓ=𝑎−𝑁+1}𝑈_ℓ𝑢₁,…,𝑢_2𝑁𝜁^ℓ=^𝑎_{ℓ=𝑎−𝑁+1}^2𝑁_𝑘=1𝑢_𝑘^{𝑁−𝑎−1}𝑈_𝑘ℓ𝑢₁,…,𝑢_𝑘−1,𝑢_𝑘+1,…,𝑢_2𝑁𝜁^ℓ=𝜁^{𝑎−𝑁+1}×^2𝑁_𝑘=1^𝑎_{ℓ=𝑎−𝑁+1}𝑈_𝑘ℓ𝑢₁,…,𝑢_𝑘−1,𝑢_𝑘+1,…,𝑢_2𝑁𝜁^{ℓ+𝑁−𝑎−1}×𝑣_𝑘^{𝑎−𝑁+1}. (255)
Next, we can see that the foregoing sum within parentheses is the expansion of the determinant D⁻_k(z,𝜁) below with respect to its 𝑘th row (by putting back the variable 𝑧): 1𝑢₁(𝑧)⋯𝑢₁^𝑁−1(𝑧)𝑢₁^{𝑏−𝑎+𝑁−1}(𝑧)⋯𝑢₁^{𝑏−𝑎+2𝑁−2}(𝑧)⋮⋮⋮⋮⋮1𝑢_𝑘−1(𝑧)⋯𝑢^𝑁−1_𝑘−1(𝑧)𝑢^{𝑏−𝑎+𝑁−1}_𝑘−1(𝑧)⋯𝑢^{𝑏−𝑎+2𝑁−2}_𝑘−1(𝑧)1𝜁⋯𝜁^𝑁−10⋯01𝑢_𝑘+1(𝑧)⋯𝑢^𝑁−1_𝑘+1(𝑧)𝑢^{𝑏−𝑎+𝑁−1}_𝑘+1(𝑧)⋯𝑢^{𝑏−𝑎+2𝑁−2}_𝑘+1(𝑧)⋮⋮⋮⋮⋮1𝑢_2𝑁(𝑧)⋯𝑢^𝑁−1_2𝑁(𝑧)𝑢^{𝑏−𝑎+𝑁−1}_2𝑁(𝑧)⋯𝑢^{𝑏−𝑎+2𝑁−2}_2𝑁(𝑧). (256)
As a result, by setting 𝐷(𝑧)=𝑈𝑢₁(𝑧),…,𝑢_2𝑁(𝑧)=1𝑢₁(𝑧)⋯𝑢₁^𝑁−1(𝑧)𝑢₁^{𝑏−𝑎+𝑁−1}(𝑧)⋯𝑢₁^{𝑏−𝑎+2𝑁−2}(𝑧)⋮⋮⋮⋮⋮1𝑢_2𝑁(𝑧)⋯𝑢^𝑁−1_2𝑁(𝑧)𝑢^{𝑏−𝑎+𝑁−1}_2𝑁(𝑧)⋯𝑢^{𝑏−𝑎+2𝑁−2}_2𝑁(𝑧), (257)
we obtain that 𝔼𝑧^𝜎𝑎𝑏𝜁^𝑆𝑎𝑏𝟙_{{𝑆𝑎𝑏≤𝑎,𝜎𝑎𝑏<+∞}}=1𝐷(𝑧)^𝑁_𝑘=1𝐷⁻_𝑘(𝑧,𝜁)𝑣_𝑘(𝑧)𝜁^{𝑎−𝑁+1}. (258)
Similarly, we could check that 𝔼𝑧^𝜎𝑎𝑏𝜁^𝑆𝑎𝑏𝟙_{{𝑆𝑎𝑏≥𝑏,𝜎𝑎𝑏<+∞}}=1𝐷(𝑧)^𝑁_𝑘=1𝐷⁺_𝑘(𝑧,𝜁)𝑣_𝑘(𝑧)𝜁^{𝑎−𝑁+1}, (259)
where 𝐷⁺_𝑘(𝑧,𝜁) is the determinant 1𝑢₁(𝑧)⋯𝑢₁^𝑁−1(𝑧)𝑢₁^{𝑏−𝑎+𝑁−1}(𝑧)⋯𝑢₁^{𝑏−𝑎+2𝑁−2}(𝑧)⋮⋮⋮⋮⋮1𝑢_𝑘−1(𝑧)⋯𝑢^𝑁−1_𝑘−1(𝑧)𝑢^{𝑏−𝑎+𝑁−1}_𝑘−1(𝑧)⋯𝑢^{𝑏−𝑎+2𝑁−2}_𝑘−1(𝑧)00⋯0𝜁^{𝑏−𝑎+𝑁−1}⋯𝜁^{𝑏−𝑎+2𝑁−2}1𝑢_𝑘+1(𝑧)⋯𝑢^𝑁−1_𝑘+1(𝑧)𝑢^{𝑏−𝑎+𝑁−1}_𝑘+1(𝑧)⋯𝑢^{𝑏−𝑎+2𝑁−2}_𝑘+1(𝑧)⋮⋮⋮⋮⋮1𝑢_2𝑁(𝑧)⋯𝑢^𝑁−1_2𝑁(𝑧)𝑢^{𝑏−𝑎+𝑁−1}_2𝑁(𝑧)⋯𝑢^{𝑏−𝑎+2𝑁−2}_2𝑁(𝑧). (260)
By adding (258) and (259) and setting 𝐷_𝑘(𝑧,𝜁)=𝐷⁺_𝑘(𝑧,𝜁)+𝐷⁻_𝑘()𝑢𝑧,𝜁=𝑈₁(𝑧),…,𝑢_𝑘−1(𝑧),𝜁,𝑢_𝑘+1(𝑧),…,𝑢_2𝑁(𝑧), (261)
we obtain that 𝔼𝑧^𝜎𝑎𝑏𝜁^𝑆𝑎𝑏𝟙_{{𝜎𝑎𝑏<+∞}}=1𝐷(𝑧)^2𝑁_𝑘=1𝐷_𝑘(𝑧,𝜁)𝑣_𝑘(𝑧)𝜁^{𝑎−𝑁+1}. (262)
We observe that the polynomials 𝐿_𝑘(𝑧,𝜁)=𝐷_𝑘(𝑧,𝜁)/𝐷(𝑧) with respect to the variable 𝜁 are of degree 𝑏−𝑎+2𝑁−2 and satisfy the equalities 𝐿_𝑘(𝑧,𝑢_𝑗(𝑧))=𝛿_𝑗𝑘 for all 𝑗∈ℰ and 𝑘∈{1,…,2𝑁}. Hence they can be expressed by means of the Lagrange fundamental polynomials as displayed in Theorem 39.

Example 40. For 𝑁=1, (248) reads 𝔼𝑧^𝜎𝑎𝑏𝜁^𝑆𝑎𝑏𝟙_{{𝜎𝑎𝑏<+∞}}=𝐿₁(𝑧,𝜁)𝑣₁(𝑧)𝜁^𝑎+𝐿₂(𝑧,𝜁)𝑣₂(𝑧)𝜁^𝑎, (263) where 𝐿₁()=𝑈𝑧,𝜁𝜁,𝑢₂(𝑧)𝑈𝑢₁(𝑧),𝑢₂(𝑧)=𝑢₂(𝑧)^𝑏−𝑎−𝜁^𝑏−𝑎𝑢₂(𝑧)^𝑏−𝑎−𝑢₁(𝑧)^𝑏−𝑎,𝐿₂()=𝑈𝑢𝑧,𝜁¹(𝑧),𝜁𝑈𝑢₁(𝑧),𝑢₂(𝑧)=𝜁^𝑏−𝑎−𝑢₁(𝑧)^𝑏−𝑎𝑢₂(𝑧)^𝑏−𝑎−𝑢₁(𝑧)^𝑏−𝑎. (264) Since 𝑣₁(𝑧)=1/𝑢₁(𝑧) and 𝑣₂(𝑧)=1/𝑢₂(𝑧), and also 𝑢₂(𝑧)=1/𝑢₁(𝑧), 𝔼𝑧^𝜎𝑎𝑏𝜁^𝑆𝑎𝑏𝟙_{{𝜎𝑎𝑏<+∞}}=𝑢₂(𝑧)^𝑏−𝑢₁(𝑧)^𝑏𝑢₂(𝑧)^𝑏−𝑎−𝑢₁(𝑧)^𝑏−𝑎𝜁^𝑎+𝑢₂(𝑧)^𝑎−𝑢₁(𝑧)^𝑎𝑢₂(𝑧)^𝑎−𝑏−𝑢₁(𝑧)^𝑎−𝑏𝜁^𝑏 (265) from which we extract the well-known formulae related to the case of an ordinary random walk: 𝔼𝑧^𝜎𝑎𝑏𝟙_{{𝑆𝑎𝑏=𝑎,𝜎𝑎𝑏<+∞}}=𝑢₂(𝑧)^𝑏−𝑢₁(𝑧)^𝑏𝑢₂(𝑧)^𝑏−𝑎−𝑢₁(𝑧)^𝑏−𝑎,𝔼𝑧^𝜎𝑎𝑏𝟙_{{𝑆𝑎𝑏=𝑏,𝜎𝑎𝑏<+∞}}=𝑢₂(𝑧)^𝑎−𝑢₁(𝑧)^𝑎𝑢₂(𝑧)^𝑎−𝑏−𝑢₁(𝑧)^𝑎−𝑏. (266) In particular, if 𝑐=1/2 (case of the classical random walk), by Example 10, we have in the above formulae that 𝑢₁(𝑧)=1−√1−𝑧²𝑧,𝑢₂(𝑧)=1+√1−𝑧²𝑧. (267) For 𝑁=2, (248) reads 𝔼𝑧^𝜎𝑎𝑏𝜁^𝑆𝑎𝑏𝟙_{{𝜎𝑎𝑏<+∞}}=𝐿₁(𝑧,𝜁)𝑣₁(𝑧)𝜁^𝑎−1+𝐿₂(𝑧,𝜁)𝑣₂(𝑧)𝜁^𝑎−1+𝐿₃(𝑧,𝜁)𝑣₃(𝑧)𝜁^𝑎−1+𝐿₄(𝑧,𝜁)𝑣₄(𝑧)𝜁^𝑎−1, (268) where 𝐿₁()=𝑈𝑧,𝜁𝜁,𝑢₂(𝑧),𝑢₃(𝑧),𝑢₄(𝑧)𝑈𝑢₁(𝑧),𝑢₂(𝑧),𝑢₃(𝑧),𝑢₄(𝑧),𝐿₂()=𝑈𝑢𝑧,𝜁¹(𝑧),𝜁,𝑢₃(𝑧),𝑢₄(𝑧)𝑈𝑢₁(𝑧),𝑢₂(𝑧),𝑢₃(𝑧),𝑢₄(𝑧),𝐿₃()=𝑈𝑢𝑧,𝜁¹(𝑧),𝑢₂(𝑧),𝜁,𝑢₄(𝑧)𝑈𝑢₁(𝑧),𝑢₂(𝑧),𝑢₃(𝑧),𝑢₄(𝑧),𝐿₄()=𝑈𝑢𝑧,𝜁¹(𝑧),𝑢₂(𝑧),𝑢₃(𝑧),𝜁𝑈𝑢₁(𝑧),𝑢₂(𝑧),𝑢₃(𝑧),𝑢₄(𝑧). (269) All the polynomials 𝐿_𝑘(𝑧,𝜁), 1≤𝑘≤4, have the form 𝐴_{𝑘,𝑎−1}(𝑧)+𝐴_𝑘,𝑎(𝑧)𝜁+𝐴_𝑘,𝑏(𝑧)𝜁^{𝑏−𝑎+1}+𝐴_𝑘,𝑏+1(𝑧)𝜁^{𝑏−𝑎+2}.

Remark 41. By expanding the determinant 𝐷_𝑘(𝑧,𝜁) with respect to its 𝑘th raw, we obtain an expansion for the polynomial 𝐿(𝑧,𝜁) as a linear combination of 1,𝜁,…,𝜁^𝑁−1,𝜁^{𝑏−𝑎+𝑁−1},𝜁^{𝑏−𝑎+𝑁},…,𝜁^{𝑏−𝑎+2𝑁−2}, that is, an expansion of the form 𝐿_𝑘(𝑧,𝜁)=^𝑁−1_ℓ=0𝐴_{𝑘,ℓ+𝑎−𝑁+1}(𝑧)𝜁^ℓ+^{𝑏−𝑎+2𝑁−2}_{ℓ=𝑏−𝑎+𝑁−1}𝐴_{𝑘,ℓ+𝑎−𝑁+1}(𝑧)𝜁^ℓ=𝜁^{𝑁−𝑎−1}_ℓ∈ℰ𝐴_𝑘ℓ(𝑧)𝜁^ℓ. (270) Hence, 𝔼𝑧^𝜎𝑎𝑏𝜁^𝑆𝑎𝑏𝟙_{{𝜎𝑎𝑏<+∞}}=^2𝑁_{𝑘=1ℓ∈ℰ}𝐴_𝑘ℓ(𝑧)𝜁^ℓ𝑣_𝑘(𝑧)^{𝑎−𝑁+1}=_ℓ∈ℰ^2𝑁_𝑘=1𝐴_𝑘ℓ(𝑧)𝑣_𝑘(𝑧)^{𝑎−𝑁+1}𝜁^ℓ (271) from which we extract, for any ℓ∈ℰ, that 𝔼𝑧^𝜎𝑎𝑏𝟙_{{𝑆𝑎𝑏=ℓ,𝜎𝑎𝑏<+∞}}=^2𝑁_𝑘=1𝐴_𝑘ℓ(𝑧)𝑢_𝑘(𝑧)^{𝑁−𝑎−1}. (272) Actually, the foregoing sum comes from the quotient 𝑈_ℓ(𝑢₁(𝑧),…,𝑢_2𝑁(𝑧))/𝑈(𝑢₁(𝑧),…,𝑢_2𝑁(𝑧)) given by (231) by expanding the determinant 𝑈_ℓ(𝑢₁(𝑧),…,𝑢_2𝑁(𝑧)) with respect to the (ℓ+𝑁−𝑎)th column or (ℓ+𝑁−𝑏+1)th column according as the number ℓ satisfies 𝑎−𝑁+1≤ℓ≤𝑎 or 𝑏≤ℓ≤𝑏+𝑁−1.

4.1.2. Pseudodistribution of 𝑆_𝑎𝑏

In order to derive the pseudodistribution of 𝑆_𝑎𝑏 which is characterized by the numbers 𝐻_𝑎𝑏,ℓ(1), ℓ∈ℰ, we solve the systems obtained by taking the limit in (262) as 𝑧→1⁻.

Lemma 42. The following identities hold: for 𝑁≤𝑘≤2𝑁−1, ^{𝑏+𝑁−1ℓ=𝑏}𝑘𝐻ℓ+𝑁−𝑎−1_𝑎𝑏,ℓ(1)=𝑘,𝑁−𝑎−1_{𝑎ℓ=𝑎−𝑁+1}𝑘𝐻𝑏+𝑁−1−ℓ_𝑎𝑏,ℓ(1)=𝑘.𝑏+𝑁−1 (273)(274)

Proof. By (85), we have the expansion 𝑢_𝑗(𝑧)=1+𝜀_𝑗(𝑧), where 𝜀_𝑗(𝑧)=_𝑧→1−𝒪(^2𝑁√1−𝑧) for any 𝑗∈{1,…,𝑁}. Actually such asymptotics holds true for any 𝑗∈{1,…,2𝑁} because of the equality 𝑢_𝑗=1/𝑢_𝑗−𝑁 for 𝑗∈{𝑁+1,…,2𝑁}. We put this into systems (240) and (241). For doing this, it is convenient to rewrite the latter as _ℓ∈ℰ𝐻_𝑎𝑏,ℓ(𝑧)𝑢_𝑗(𝑧)^{ℓ+𝑁−𝑎−1}=𝑢_𝑗(𝑧)^{𝑁−𝑎−1},1≤𝑗≤2𝑁,_ℓ∈ℰ𝐻_𝑎𝑏,ℓ(𝑧)𝑢_𝑗(𝑧)^{𝑏+𝑁−1−ℓ}=𝑢_𝑗(𝑧)^{𝑏+𝑁−1},1≤𝑗≤2𝑁. (275) We obtain that _ℓ∈ℰ1+𝜀_𝑗(𝑧)^{ℓ+𝑁−𝑎−1}𝐻_𝑎𝑏,ℓ(𝑧)=1+𝜀_𝑗(𝑧)^{𝑁−𝑎−1},1≤𝑗≤2𝑁,_ℓ∈ℰ1+𝜀_𝑗(𝑧)^{𝑏+𝑁−1−ℓ}𝐻_𝑎𝑏,ℓ(𝑧)=1+𝜀_𝑗(𝑧)^{𝑏+𝑁−1},1≤𝑗≤2𝑁. (276)(277) System (276) writes ^{𝑏−𝑎+2𝑁−2𝑘=0ℓ∈ℰ∶}_{ℓ≥𝑘+𝑎−𝑁+1}𝑘𝐻ℓ+𝑁−𝑎−1_𝑎𝑏,ℓ(𝑧)𝜀_𝑗(𝑧)^𝑘=^{𝑁−𝑎−1}_𝑘=0𝑘𝜀𝑁−𝑎−1_𝑗(𝑧)^𝑘. (278) Set 𝑀_𝑘(𝑧)=_{ℓ∈ℰ∶ℓ≥𝑘+𝑎−𝑁+1}𝑘𝐻ℓ+𝑁−𝑎−1_𝑎𝑏,ℓ(𝑧)−𝑘,𝑅𝑁−𝑎−1_𝑗(𝑧)=−^{𝑏−𝑎+2𝑁−2}_𝑘=2𝑁𝑀_𝑘(𝑧)𝜀_𝑗(𝑧)^𝑘. (279) Then, equality (278) reads ^2𝑁−1_𝑘=0𝑀_𝑘(𝑧)𝜀_𝑗(𝑧)^𝑘=𝑅_𝑗(𝑧),1≤𝑗≤2𝑁. (280) This is a Vandermonde system which can be solved as in the proof of Lemma 21 upon changing 𝑁 into 2𝑁. We can check that lim_𝑧→1⁻𝑀_𝑘(𝑧)=0, which entails that ^ℓ∈ℰ∶_{ℓ≥𝑘+𝑎−𝑁+1}𝑘𝐻ℓ+𝑁−𝑎−1_𝑎𝑏,ℓ(1)=𝑘𝑁−𝑎−1,0≤𝑘≤2𝑁−1. (281) Similarly, using (277), we can prove that ^ℓ∈ℰ∶_{ℓ≤𝑏+𝑁−1−𝑘}𝑘𝐻𝑏+𝑁−ℓ−1_𝑎𝑏,ℓ(1)=𝑘𝑁+𝑏−1,0≤𝑗≤2𝑁−1. (282)
Actually, we will only use (281) and (282) restricted to 𝑗∈{𝑁,…,2𝑁−1} which immediately yields system (273) and (274).

Now, we state one of the most important result of this work. We solve the famous problem of the “gambler’s ruin” in the context of the pseudorandom walk.

Theorem 43. The pseudodistribution of 𝑆_𝑎𝑏 is given, for ℓ∈{0,1,…,𝑁−1}, by ℙ𝑆_𝑎𝑏=𝑎−ℓ,𝜎_𝑎𝑏=()<+∞−1^ℓ(𝐾𝑁/(ℓ−𝑎))_ℓ^𝑁−1_{𝑁ℓ+𝑏−𝑎+𝑁−1},ℙ𝑆_𝑎𝑏=𝑏+ℓ,𝜎_𝑎𝑏=()<+∞−1^ℓ(𝐾𝑁/(ℓ+𝑏))_ℓ^𝑁−1_{𝑁ℓ+𝑏−𝑎+𝑁−1}, (283) where 𝑁𝑁=()𝐾=𝑁−𝑎−1𝑁+𝑏−1−1^𝑁𝑎()⋯()𝑏()⋯().𝑁!𝑎−1𝑎−𝑁+1𝑏+1𝑏+𝑁−1 (284) Moreover, ℙ{𝜎_𝑎𝑏<+∞}=1 and ℙ𝜎⁺_𝑏<𝜎⁻_𝑎=𝐾𝑁²_𝒟⁺𝑢^−𝑎−1(1−𝑢)^𝑁−1𝑣^𝑏−1(1−𝑣)^𝑁−1d𝑢dℙ𝜎𝑣,_−𝑎<𝜎⁺_𝑏=𝐾𝑁²_𝒟⁻𝑢^−𝑎−1(1−𝑢)^𝑁−1𝑣^𝑏−1(1−𝑣)^𝑁−1d𝑢d𝑣, (285) where 𝒟⁺=(𝑢,𝑣)∈ℝ²,𝒟∶0≤𝑣≤𝑢≤1₋=(𝑢,𝑣)∈ℝ²∶0≤𝑢≤𝑣≤1. (286)

Proof. We have to solve system (273) and (274). For (273), for instance, the principal matrix and the right-hand side matrix are ℓ+𝑏−𝑎+𝑁−1𝑘+𝑁_{0≤𝑘,ℓ≤𝑁−1},𝑁−𝑎−1𝑘+𝑁_{0≤𝑘≤𝑁−1} (287) and the matrix form of the solution is given by ℓ+𝑏−𝑎+𝑁−1𝑘+𝑁⁻¹_{0≤𝑘,ℓ≤𝑁−1}𝑁−𝑎−1𝑘+𝑁^{0≤𝑘≤𝑁−1}. (288)
The computation of this product being quite fastidious, we postponed it to Appendix A.3. The result is given by Theorem A.8: (−1)^ℓ(𝐾𝑁/(ℓ+𝑏))_ℓ^𝑁−1_{𝑁ℓ+𝑏−𝑎+𝑁−10≤ℓ≤𝑁−1}. (289)
The entries of this matrix provide the pseudo-probabilities ℙ{𝑆_𝑎𝑏=𝑏+ℓ,𝜎_𝑎𝑏<+∞}, 0≤ℓ≤𝑁−1, which are exhibited in Theorem 43. The analogous formula for ℙ{𝑆_𝑎𝑏=𝑎−ℓ,𝜎_𝑎𝑏<+∞} holds true in the same way.
Next, by observing that 𝜎_𝑎𝑏=min(𝜎⁻_𝑎,𝜎⁺_𝑏) and that {𝜎_𝑎𝑏<+∞}={𝜎⁻_𝑎<+∞}∪{𝜎⁺_𝑏<+∞}, we have that ℙ𝜎⁺_𝑏<𝜎⁻_𝑎=ℙ𝜎⁺_𝑏<𝜎⁻_𝑎,𝜎_𝑎𝑏<+∞=ℙ𝑆_𝑎𝑏≥𝑏,𝜎_𝑎𝑏<+∞=^𝑁−1_ℓ=0ℙ𝑆_𝑎𝑏=𝑏+ℓ,𝜎_𝑎𝑏<+∞=𝐾^𝑁−1_ℓ=0(−1)^ℓ(𝑁/(ℓ+𝑏))_ℓ^𝑁−1_{𝑁ℓ+𝑏−𝑎+𝑁−1}. (290)(291) Noticing that 1/(ℓ+𝑏)=∫¹₀𝑦^ℓ+𝑏−1d𝑦 and 1/_𝑁^{ℓ+𝑏−𝑎+𝑁−1}=𝑁∫¹₀𝑥^{ℓ+𝑏−𝑎−1}(1−𝑥)^𝑁−1d𝑥, we get that ^𝑁−1_ℓ=0(−1)^ℓ(𝑁/(ℓ+𝑏))_ℓ^𝑁−1_{𝑁ℓ+𝑏−𝑎+𝑁−1}=𝑁²¹₀¹₀^𝑁−1_ℓ=0(−1)^ℓℓ𝑁−1𝑥𝑦^ℓ×𝑥^{𝑏−𝑎−1}(1−𝑥)^𝑁−1𝑦^𝑏−1d𝑥d𝑦=𝑁²¹₀¹₀𝑥^{𝑏−𝑎−1}(1−𝑥)^𝑁−1𝑦^𝑏−11−𝑥𝑦^𝑁−1d𝑥d𝑦. (292) The computations can be pursued by performing the change of variables (𝑢,𝑣)=(𝑥,𝑥𝑦) in the above integral: ^𝑁−1_ℓ=0(−1)^ℓ(𝑁/(ℓ+𝑏))_ℓ^𝑁−1_{𝑁ℓ+𝑏−𝑎+𝑁−1}=𝑁²_𝒟⁺𝑢^−𝑎−1(1−𝑢)^𝑁−1𝑣^𝑏−1(1−𝑣)^𝑁−1d𝑢d𝑣. (293) Putting (293) into (291) yields the expression of ℙ{𝜎⁺_𝑏<𝜎⁻_𝑎} displayed in Theorem 43. The similar expression for ℙ{𝜎⁻_𝑎<𝜎⁺_𝑏} holds true. Finally, ℙ𝜎_𝑎𝑏𝜎<+∞=ℙ⁻_𝑎<𝜎⁺_𝑏,𝜎_𝑎𝑏<+∞+ℙ𝜎⁺_𝑏<𝜎⁻_𝑎,𝜎_𝑎𝑏𝜎<+∞=ℙ⁻_𝑎<𝜎⁺_𝑏+ℙ𝜎⁺_𝑏<𝜎⁻_𝑎=𝐾𝑁²¹₀¹₀𝑢^−𝑎−1(1−𝑢)^𝑁−1𝑣^𝑏−1(1−𝑣)^𝑁−1d𝑢d𝑣. (294) The foregoing integral is quite elementary: ¹₀¹₀𝑢^−𝑎−1(1−𝑢)^𝑁−1𝑣^𝑏−1(1−𝑣)^𝑁−1d𝑢d𝑣=¹₀𝑢^−𝑎−1(1−𝑢)^𝑁−1d𝑢¹₀𝑣^𝑏−1(1−𝑣)^𝑁−1d𝑣()𝐵()=1=𝐵𝑁,−𝑎𝑁,𝑏𝑁²_𝑁^{𝑁−𝑎−1}_𝑁^{𝑏+𝑁−1}=1𝐾𝑁², (295) which entails that ℙ{𝜎_𝑎𝑏<+∞}=1.

In the sequel, when considering 𝑆_𝑎𝑏, we will omit the condition 𝜎_𝑎𝑏<+∞.

Example 44. Let us have a look on the particular values 1,2,3 of 𝑁.
(i) Case 𝑁=1. In this case 𝑆_𝑎𝑏∈{𝑎,𝑏} and ℙ𝑆_𝑎𝑏=𝑎=ℙ𝜎⁻_𝑎<𝜎⁺_𝑏=𝑏,ℙ𝑆𝑏−𝑎_𝑎𝑏=𝑏=ℙ𝜎⁺_𝑏<𝜎⁻_𝑎𝑎=−.𝑏−𝑎 (296) We retrieve one of the most well-known and important results for the ordinary random walk: this is the famous problem of the gambler’s ruin!
(ii) Case  𝑁=2. In this case, the pseudorandom variables 𝜉_𝑛, 𝑛∈ℕ^∗, have two-valued upward jumps and two-valued downward jumps. Hence, the exit place must be either 𝑎−1, 𝑎, 𝑏 or 𝑏+1: 𝑆_𝑎𝑏∈{𝑎−1,𝑎,𝑏,𝑏+1}. We have that ℙ𝑆_𝑎𝑏=()=𝑎−1𝑎𝑏𝑏+1(),ℙ𝑆𝑏−𝑎+1)(𝑏−𝑎+2_𝑎𝑏()𝑏()=𝑎=−𝑎−1𝑏+1(),ℙ𝑆𝑏−𝑎)(𝑏−𝑎+1_𝑎𝑏=𝑎()=𝑏𝑎−1)(𝑏+1(),ℙ𝑆𝑏−𝑎)(𝑏−𝑎+1_𝑎𝑏𝑎()𝑏=𝑏+1=−𝑎−1(),ℙ𝜎𝑏−𝑎+1)(𝑏−𝑎+2_−𝑎<𝜎⁺_𝑏=𝑏(𝑏+1)(𝑏−3𝑎+2)(),ℙ𝜎𝑏−𝑎)(𝑏−𝑎+1)(𝑏−𝑎+2_+𝑏<𝜎⁻_𝑎=𝑎(𝑎−1)(3𝑏−𝑎+2)().𝑏−𝑎)(𝑏−𝑎+1)(𝑏−𝑎+2 (297) We can easily check that ℙ{𝑆_𝑎𝑏=𝑎}+ℙ{𝑆_𝑎𝑏=𝑎−1}+ℙ{𝑆_𝑎𝑏=𝑏}+ℙ{𝑆_𝑎𝑏=𝑏+1}=1 as well as ℙ{𝜎⁻_𝑎<𝜎⁺_𝑏}+ℙ{𝜎⁺_𝑏<𝜎⁻_𝑎}=1.
(iii) Case 𝑁=3. In this case 𝑆_𝑎𝑏∈{𝑎−2,𝑎−1,𝑎,𝑏,𝑏+1,𝑏+2} and ℙ𝑆_𝑎𝑏=𝑎()𝑏()=𝑎−2𝑎−1𝑏+1)(𝑏+2(),ℙ𝑆𝑏−𝑎+2)(𝑏−𝑎+3)(𝑏−𝑎+4_𝑎𝑏𝑎()𝑏()=𝑎−1=−𝑎−2𝑏+1)(𝑏+2(),ℙ𝑆𝑏−𝑎+1)(𝑏−𝑎+2)(𝑏−𝑎+3_𝑎𝑏=()𝑏()=𝑎𝑎−1)(𝑎−2𝑏+1)(𝑏+2(),ℙ𝑆𝑏−𝑎)(𝑏−𝑎+1)(𝑏−𝑎+2_𝑎𝑏𝑎()=𝑏=−𝑎−1)(𝑎−2)(𝑏+1)(𝑏+2(),ℙ𝑆𝑏−𝑎)(𝑏−𝑎+1)(𝑏−𝑎+2_𝑎𝑏=𝑎()𝑏()=𝑏+1𝑎−1)(𝑎−2𝑏+2(),ℙ𝑆𝑏−𝑎+1)(𝑏−𝑎+2)(𝑏−𝑎+3_𝑎𝑏𝑎()𝑏()=𝑏+2=−𝑎−1)(𝑎−2𝑏+1(),ℙ𝜎𝑏−𝑎+2)(𝑏−𝑎+3)(𝑏−𝑎+4_−𝑏<𝜎⁺_𝑎=𝑏(𝑏+1)(𝑏+2)10𝑎²−5𝑎𝑏+𝑏²−25𝑎+7𝑏+12(),ℙ𝜎𝑏−𝑎)(𝑏−𝑎+1)(𝑏−𝑎+2)(𝑏−𝑎+3)(𝑏−𝑎+4_+𝑏<𝜎⁻_𝑎𝑎()𝑎=−𝑎−1)(𝑎−2²−5𝑎𝑏+10𝑏²−7𝑎+25𝑏+12().𝑏−𝑎)(𝑏−𝑎+1)(𝑏−𝑎+2)(𝑏−𝑎+3)(𝑏−𝑎+4 (298)

4.1.3. Pseudomoments of 𝑆_𝑎𝑏

Let us recall the notation we previously introduced in Section 3.1.3: (𝑖)_𝑛=𝑖(𝑖−1)(𝑖−2)⋯(𝑖−𝑛+1) for any 𝑖∈ℤ and any 𝑛∈ℕ^∗ and (𝑖)₀=1, as well as the conventions 1/𝑖!=0 for any negative integer 𝑖 and ∑^𝑗_𝑘=𝑖=0 if 𝑖>𝑗.

In this section, we compute several functionals related to the pseudomoments of 𝑆_𝑎𝑏. Namely, we provide formulae for 𝔼[𝑆_𝑎𝑏(𝑆_𝑎𝑏−𝑏)_𝑛−1] (Theorem 46), 𝔼[(𝑆_𝑎𝑏)^𝑛] (Corollary 47), and 𝔼[(𝑆_𝑎𝑏−𝑏)_𝑛] (Theorem 48). This schedule may seem surprising; actually, we have been able to carry out the calculations by following this chronology.

Putting the identities 1/_𝑁^{ℓ+𝑏−𝑎+𝑁−1}=𝑁∫¹₀𝑥^{ℓ+𝑏−𝑎−1}(1−𝑥)^𝑁−1d𝑥 and 1/(ℓ+𝑏)=∫¹₀𝑦^ℓ+𝑏−1d𝑦 into the equality 𝔼𝑓𝑆_𝑎𝑏𝟙_{{𝑆𝑎𝑏≥𝑏}}=𝐾𝑁^𝑁−1_ℓ=0(−1)^ℓ𝑓(ℓ+𝑏)/(ℓ+𝑏_ℓ^𝑁−1_{𝑁ℓ+𝑏−𝑎+𝑁−1}), (299) we immediately get the following integral representations for 𝔼[𝑓(𝑆_𝑎𝑏)𝟙_{{𝑆𝑎𝑏≥𝑏}}], and the analogous ones hold true for 𝔼[𝑓(𝑆_𝑎𝑏)𝟙_{{𝑆𝑎𝑏≤𝑎}}].

Theorem 45. For any function 𝑓 defined on ℰ, 𝔼𝑓𝑆_𝑎𝑏𝟙_{{𝑆𝑎𝑏≥𝑏}}=𝐾𝑁²¹₀^𝑁−1_ℓ=0(−1)^ℓℓ𝑁−1𝑓(ℓ+𝑏)𝑥ℓ+𝑏^ℓ×𝑥^{𝑏−𝑎−1}(1−𝑥)^𝑁−1d𝑥=𝐾𝑁²¹₀¹₀^𝑁−1_ℓ=0(−1)^ℓℓ𝑓()𝑁−1ℓ+𝑏𝑥𝑦^ℓ×𝑥^{𝑏−𝑎−1}(1−𝑥)^𝑁−1𝑦^𝑏−1d𝑥d𝑦,𝔼𝑓𝑆_𝑎𝑏𝟙_{{𝑆𝑎𝑏≥𝑎}}=𝐾𝑁²¹₀^𝑁−1_ℓ=0(−1)^ℓℓ𝑁−1𝑓(𝑎−ℓ)𝑥ℓ−𝑎^ℓ×𝑥^{𝑏−𝑎−1}(1−𝑥)^𝑁−1d𝑥=𝐾𝑁²¹₀¹₀^𝑁−1_ℓ=0(−1)^ℓℓ𝑓()𝑁−1𝑎−ℓ𝑥𝑦^ℓ×𝑥^{𝑏−𝑎−1}(1−𝑥)^𝑁−1𝑦^−𝑎−1d𝑥d𝑦. (300)(301)(302)(303)

In view of (300) and (302) and in order to compute the pseudomoments of 𝑆_𝑎𝑏, it is convenient to introduce the function 𝑓_𝑛 defined by 𝑓_𝑛(𝑖)=𝑖(𝑖−𝑏)_𝑛−1 for any integers 𝑖 and 𝑛 such that 𝑛≥1. In particular, 𝑓₁(𝑖)=𝑖. We immediately see that, by choosing 𝑓=𝑓₁ in Theorem 45, quantities (300) and (302) are opposite. As a byproduct, 𝔼[𝑆_𝑎𝑏]=0. More generally, we have the results below.

Theorem 46. For any positive integer 𝑛, 𝔼𝑆_𝑎𝑏𝑆_𝑎𝑏−𝑏_𝑛−1𝟙_{{𝑆𝑎𝑏≥𝑏}}=(−1)^𝑛−1(𝐾𝑁/(2𝑁−𝑛))(𝑁!/(𝑁−𝑛_{2𝑁+𝑏−𝑎−22𝑁−𝑛})!)𝑖𝑓1≤𝑛≤𝑁,0𝑖𝑓𝑛≥𝑁+1,𝔼𝑆_𝑎𝑏𝑆_𝑎𝑏−𝑏_𝑛−1𝟙_{{𝑆𝑎𝑏≤𝑎}}=(−1)^𝑛(𝐾𝑁/(2𝑁−𝑛))(𝑁!/(𝑁−𝑛_{2𝑁+𝑏−𝑎−22𝑁−𝑛})!)()𝑖𝑓1≤𝑛≤𝑁,0𝑖𝑓𝑁+1≤𝑛≤2𝑁−1,−1^{𝑁+𝑛−1}𝐾𝑁𝑁!(𝑛−𝑁−1)!𝑛+𝑏−𝑎−2𝑛−2𝑁𝑖𝑓𝑛≥2𝑁. (304)(305) In particular, for any 𝑛∈{1,…,2𝑁−1}, 𝔼𝑆_𝑎𝑏𝑆_𝑎𝑏−𝑏_𝑛−1=0. (306)

Proof. By (300), we get that 𝔼𝑓_𝑛𝑆_𝑎𝑏𝟙_{{𝑆𝑎𝑏≥𝑏}}=𝐾𝑁²¹₀^𝑁−1_ℓ=0(−1)^ℓℓ(ℓ)𝑁−1_𝑛−1𝑥^{ℓ+𝑏−𝑎−1}×(1−𝑥)^𝑁−1d𝑥. (307) By noticing that (ℓ)_𝑛−1=0 for ℓ∈{0,…,𝑛−2} and _ℓ^𝑁−1(ℓ)_𝑛−1=^𝑁−𝑛_ℓ−𝑛+1(𝑁−1)_𝑛−1 for ℓ≥𝑛−1, we obtain that ^𝑁−1_ℓ=0(−1)^ℓℓ(ℓ)𝑁−1_𝑛−1𝑥^ℓ=(𝑁−1)_𝑛−1𝟙_{{1≤𝑛≤𝑁}}^𝑁−1_ℓ=𝑛−1(−1)^ℓ𝑥𝑁−𝑛ℓ−𝑛+1^ℓ=(𝑁−1)_𝑛−1𝟙_{{1≤𝑛≤𝑁}}𝑥^{𝑛−1𝑁−𝑛}_ℓ=0(−1)^ℓ+𝑛−1ℓ𝑥𝑁−𝑛_ℓ=(−1)^𝑛−1(𝑁−1)!()!𝑥𝑁−𝑛^𝑛−1(1−𝑥)^𝑁−𝑛if01≤𝑛≤𝑁,if𝑛≥𝑁+1. (308) Hence, if 1≤𝑛≤𝑁, 𝔼𝑓_𝑛𝑆_𝑎𝑏𝟙_{{𝑆𝑎𝑏≥𝑏}}=(−1)^𝑛−1𝐾𝑁²(𝑁−1)!(𝑁−𝑛)!¹₀𝑥^{𝑛+𝑏−𝑎−2}(1−𝑥)^{2𝑁−𝑛−1}d𝑥=(−1)^𝑛−1𝐾𝑁𝑁!()!()!()!𝑁−𝑛𝑛+𝑏−𝑎−22𝑁−𝑛−1(2𝑁+𝑏−𝑎−2)! (309)
and we arrive at (304). Moreover, if 𝑛≥𝑁+1 and 𝑆_𝑎𝑏≥𝑏, we have 𝑆_𝑎𝑏∈{𝑏,𝑏+1,…,𝑏+𝑁−1}. Then, it is clear that 𝑓_𝑛(𝑆_𝑎𝑏)=0 and (304) still holds in this case.
On the other hand, by Theorem 43, we get that 𝔼𝑓_𝑛𝑆_𝑎𝑏𝟙_{{𝑆𝑎𝑏≤𝑎}}=^𝑁−1_ℓ=0ℙ𝑆_𝑎𝑏=𝑎−ℓ𝑓_𝑛(𝑎−ℓ)=𝐾𝑁^𝑁−1_ℓ=0(−1)^ℓ−1(𝑎−𝑏−ℓ)_𝑛−1ℓ^𝑁−1_{𝑁ℓ+𝑏−𝑎+𝑁−1}=(−1)^𝑛𝐾𝑁𝑁!^𝑁−1_ℓ=0(−1)^ℓℓ𝑁−1(ℓ+𝑏−𝑎+𝑛−2)!()!.ℓ+𝑏−𝑎+𝑁−1 (310) For 𝑛≤𝑁, we can write that (ℓ+𝑏−𝑎+𝑛−2)!()!=1ℓ+𝑏−𝑎+𝑁−1()!1𝑁−𝑛(ℓ+𝑏−𝑎+𝑁−1)^{ℓ+𝑏−𝑎+𝑁−2}_𝑁−𝑛=1(𝑁−𝑛)!¹₀𝑥^{ℓ+𝑛+𝑏−𝑎−2}(1−𝑥)^𝑁−𝑛d𝑥. (311) Then, 𝔼𝑓_𝑛𝑆_𝑎𝑏𝟙_{{𝑆𝑎𝑏≤𝑎}}=(−1)^𝑛𝐾𝑁𝑁!(𝑁−𝑛)!×¹₀^𝑁−1_ℓ=0(−1)^ℓℓ𝑥𝑁−1_{ℓ+𝑛+𝑏−𝑎−2}(1−𝑥)^𝑁−𝑛d𝑥=(−1)^𝑛𝐾𝑁𝑁!(𝑁−𝑛)!¹₀𝑥^{𝑛+𝑏−𝑎−2}(1−𝑥)^{2𝑁−𝑛−1}d𝑥=(−1)^𝑛𝐾𝑁𝑁!()!×()!()!𝑁−𝑛𝑛+𝑏−𝑎−22𝑁−𝑛−1(2𝑁+𝑏−𝑎−2)! (312) which proves (305). For 𝑛≥𝑁+1, we write instead that (ℓ+𝑏−𝑎+𝑛−2)!()!=ℓ+𝑏−𝑎+𝑁−1d^{𝑛−𝑁−1}d𝑥^{𝑛−𝑁−1}𝑥^{ℓ+𝑛+𝑏−𝑎−2}_𝑥=1. (313) Therefore, 𝔼𝑓_𝑛𝑆_𝑎𝑏𝟙_{{𝑆𝑎𝑏≤𝑎}}=(−1)^𝑛×𝐾𝑁𝑁!d^{𝑛−𝑁−1}d𝑥^{𝑛−𝑁−1𝑁−1}_ℓ=0(−1)^ℓℓ𝑥𝑁−1_{ℓ+𝑛+𝑏−𝑎−2𝑥=1}=(−1)^𝑛𝐾𝑁𝑁!d^{𝑛−𝑁−1}d𝑥^{𝑛−𝑁−1}𝑥^{𝑛+𝑏−𝑎−2}(1−𝑥)^𝑁−1_𝑥=1. (314) If 𝑁+1≤𝑛≤2𝑁−1, the above derivative vanishes since 1 is a root of multiplicity 𝑁−1 of the polynomial 𝑥^{ℓ+𝑛+𝑏−𝑎−2}(1−𝑥)^𝑁−1. Finally, for 𝑛≥2𝑁, we appeal to Leibniz rule for evaluating the derivative of interest: d^{𝑛−𝑁−1}d𝑥^{𝑛−𝑁−1}𝑥^{𝑛+𝑏−𝑎−2}(1−𝑥)^𝑁−1_𝑥=1=^{𝑛−𝑁−1}_𝑘=0(−1)^𝑘𝑘×()!𝑛−𝑁−1𝑛+𝑏−𝑎−2()!()𝑘+𝑁+𝑏−𝑎−1𝑁−1_𝑘𝛿_{𝑘,𝑁−1}=(−1)^𝑁−1()!()!𝑁−1𝑛+𝑏−𝑎−2()!=()2𝑁+𝑏−𝑎−2𝑛−𝑁−1𝑁−1−1^𝑁−1(𝑛−𝑁−1)!.𝑛+𝑏−𝑎−2𝑛−2𝑁 (315) This proves (305) in this case.

Corollary 47. For 𝑛∈{1,…,2𝑁−1}, the pseudo-moment of 𝑆_𝑎𝑏 of order 𝑛 vanishes: 𝔼𝑆_𝑎𝑏^𝑛=0. (316) Moreover, 𝔼𝑆_𝑎𝑏^2𝑁=−𝑎(𝑎−1)⋯(𝑎−𝑁+1)𝑏(𝑏+1)⋯(𝑏+𝑁−1). (317)

Proof. As in the proof of Theorem 27, we appeal to the following argument: the polynomial 𝑋^𝑛 is a linear combination of 𝑓₁(𝑋),…,𝑓_𝑛(𝑋). Then 𝔼[(𝑆_𝑎𝑏)^𝑛] can be written as a linear combination of 𝔼[𝑓₁(𝑆_𝑎𝑏)],…,𝔼[𝑓_𝑛(𝑆_𝑎𝑏)] which vanish when 1≤𝑛≤2𝑁−1. Thus, 𝔼[(𝑆_𝑎𝑏)^𝑛]=0. The same argument entails that 𝔼𝑆_𝑎𝑏^2𝑁=𝔼𝑓_2𝑁𝑆_𝑎𝑏=(−1)^𝑁−1𝐾𝑁!² (318)
which proves (317).

In the following theorem, we provide an integral representation for certain factorial pseudomoments of (𝑆_𝑎𝑏−𝑎) and (𝑆_𝑎𝑏−𝑏) which will be used in the next section.

Theorem 48. For any integer 𝑛∈{0,…,𝑁−1}, 𝔼𝑆_𝑎𝑏−𝑏_𝑛𝟙_{{𝑆𝑎𝑏≥𝑏}}=(−1)^𝑛𝐾𝑁²(𝑁−1)_𝑛×_𝒟⁺𝑢^−𝑎−1(1−𝑢)^𝑁−1𝑣^{𝑛+𝑏−1}(1−𝑣)^{𝑁−𝑛−1}d𝑢d𝑣,𝔼𝑎−𝑆_𝑎𝑏𝑛𝟙_{{𝑆𝑎𝑏≤𝑎}}=(−1)^𝑛𝐾𝑁²(𝑁−1)_𝑛×_𝒟⁻𝑢^{𝑛−𝑎−1}(1−𝑢)^{𝑁−𝑛−1}𝑣^𝑏−1(1−𝑣)^𝑁−1d𝑢d𝑣. (319)(320) The above identities can be rewritten as 𝔼𝑆_𝑎𝑏𝑛𝟙−𝑏_{{𝑆𝑎𝑏≥𝑏}}=(−1)^𝑛𝐾𝑁²𝑛×𝑁−1_𝒟+𝑢^−𝑎−1(1−𝑢)^𝑁−1𝑣^{𝑛+𝑏−1}(1−𝑣)^{𝑁−𝑛−1}d𝑢d𝑣,𝔼𝑎−𝑆_𝑎𝑏𝑛𝟙_{{𝑆𝑎𝑏≤𝑎}}=(−1)^𝑛𝐾𝑁²𝑛×𝑁−1_𝒟−𝑢^{𝑛−𝑎−1}(1−𝑢)^{𝑁−𝑛−1}𝑣^𝑏−1(1−𝑣)^𝑁−1d𝑢d𝑣. (321)(322)

Proof. By (301), we have that 𝔼𝑆_𝑎𝑏−𝑏_𝑛𝟙_{{𝑆𝑎𝑏≥𝑏}}=𝐾𝑁²¹₀¹₀^𝑁−1_ℓ=0(−1)^ℓℓ(ℓ)𝑁−1_𝑛𝑥𝑦^ℓ×𝑥^{𝑏−𝑎−1}(1−𝑥)^𝑁−1𝑦^𝑏−1d𝑥d𝑦. (323) The sum lying in the above integral can be easily calculated: ^𝑁−1_ℓ=0(−1)^ℓℓ(ℓ)𝑁−1_𝑛𝑥𝑦^ℓ=(𝑁−1)_𝑛^𝑁−1_ℓ=𝑛(−1)^ℓ𝑁−𝑛−1ℓ−𝑛𝑥𝑦^ℓ=(−1)^𝑛(𝑁−1)_𝑛𝑥𝑦^𝑛1−𝑥𝑦^{𝑁−𝑛−1}. (324) Hence, 𝔼𝑆_𝑎𝑏−𝑏_𝑛𝟙_{{𝑆𝑎𝑏≥𝑏}}=(−1)^𝑛(𝑁−1)_𝑛𝐾𝑁²×¹₀¹₀𝑥^{𝑛+𝑏−𝑎−1}(1−𝑥)^𝑁−1𝑦^{𝑛+𝑏−1}1−𝑥𝑦^{𝑁−𝑛−1}d𝑥d𝑦. (325) Performing the change of variables (𝑢,𝑣)=(𝑥,𝑥𝑦) in the foregoing integral immediately yields (319). Formula (320) can be deduced from (303) exactly in the same way.

4.2. Link with High-Order Finite-Difference Equations

Set Δ⁺𝑓(𝑖)=𝑓(𝑖+1)−𝑓(𝑖) and Δ⁻𝑓(𝑖)=𝑓(𝑖)−𝑓(𝑖−1) for any 𝑖∈ℤ and (Δ⁺)^𝑗=Δ⁺∘⋯∘Δ⁺_𝑗times and (Δ⁻)^𝑗=Δ⁻∘⋯∘Δ⁻_𝑗times for any 𝑗∈ℕ^∗. Set also (Δ⁺)⁰𝑓=(Δ⁻)⁰𝑓=𝑓. The quantities (Δ⁺)^𝑗 and (Δ⁻)^𝑗 are the iterated forward and backward finite-difference operators given by Δ^+𝑗𝑓(𝑖)=^𝑗_𝑘=0(−1)^𝑗+𝑘𝑗𝑘𝑓(),(Δ𝑖+𝑘₋)^𝑗𝑓(𝑖)=^𝑗_𝑘=0(−1)^𝑘𝑗𝑘𝑓(𝑖−𝑘). (326) Conversely, 𝑓(𝑖+𝑘) and 𝑓(𝑖−𝑘) can be expressed by means of (Δ⁺)^𝑗𝑓(𝑖),(Δ⁻)^𝑗𝑓(𝑖), 0≤𝑗≤𝑘, according to 𝑓(𝑖+𝑘)=^𝑘_𝑗=0𝑘𝑗Δ^+𝑗𝑓(𝑖),𝑓(𝑖−𝑘)=^𝑘_𝑗=0(−1)^𝑗𝑘𝑗(Δ⁻)^𝑗𝑓(𝑖). (327) We have the following expression for any functional of the pseudorandom variable 𝑆_𝑎𝑏.

Theorem 49. One has, for any function 𝑓 defined on ℰ, that 𝔼𝑓𝑆_𝑎𝑏=^𝑁−1_𝑗=0𝐼⁻_{𝑎𝑏,𝑗}(Δ⁻)^𝑗𝑓(𝑎)+^𝑁−1_𝑗=0𝐼⁺_{𝑎𝑏,𝑗}Δ^+𝑗𝑓(𝑏) (328) with 𝐼⁻_{𝑎𝑏,𝑗}=𝐾𝑁²𝑗×𝑁−1_𝒟−𝑢^{𝑗−𝑎−1}(1−𝑢)^{𝑁−𝑗−1}𝑣^𝑏−1(1−𝑣)^𝑁−1d𝑢d𝐼𝑣,_{+𝑎𝑏,𝑗}=(−1)^𝑗𝐾𝑁²𝑗𝑁−1×_𝒟⁺𝑢^−𝑎−1(1−𝑢)^𝑁−1𝑣^{𝑗+𝑏−1}(1−𝑣)^{𝑁−𝑗−1}d𝑢d𝑣. (329)

Proof. By (327), we see that 𝔼𝑓𝑆_𝑎𝑏=𝔼𝟙_{{𝑆𝑎𝑏≤𝑎}}^{𝑎−𝑆𝑎𝑏}_𝑗=0(−1)^𝑗𝑎−𝑆_𝑎𝑏𝑗(Δ⁻)^𝑗𝑓(𝑎)+𝔼𝟙_{{𝑆𝑎𝑏}^𝑆_≥𝑏}^{𝑎𝑏−𝑏}_𝑗=0𝑆_𝑎𝑏𝑗Δ−𝑏_+𝑗𝑓(𝑏)=^𝑁−1_𝑗=0(−1)^𝑗𝔼𝟙_{{𝑆𝑎𝑏≤𝑎}}𝑎−𝑆_𝑎𝑏𝑗(Δ⁻)^𝑗𝑓(𝑎)+^𝑁−1_𝑗=0𝔼𝟙_{{𝑆𝑎𝑏≥𝑏}}𝑆_𝑎𝑏𝑗Δ−𝑏_+𝑗𝑓(𝑏) (330) which immediately yields (328) thanks to (321) and (322).

Corollary 50. The generating function of 𝑆_𝑎𝑏 is given by 𝔼𝜁^𝑆𝑎𝑏=𝜁^𝑎𝑁−1_𝑗=0𝐼⁻_{𝑎𝑏,𝑗}11−𝜁^𝑗+𝜁^𝑏𝑁−1_𝑗=0𝐼⁺_{𝑎𝑏,𝑗}(𝜁−1)^𝑗. (331)

Proof. Let us apply Theorem 49 to the function 𝑓(𝑖)=𝜁^𝑖 for which we plainly have (Δ⁺)^𝑗𝑓(𝑖)=𝜁^𝑖(𝜁−1)^𝑗 and (Δ⁻)^𝑗𝑓(𝑖)=𝜁^𝑖(1−1/𝜁)^𝑗. This immediately yields (331).

Of special interest is the case when the starting point of the pseudorandom walk is some point 𝑥∈ℤ. By translating 𝑎, 𝑏 into 𝑎−𝑥, 𝑏−𝑥 and the function 𝑓 into the shifted function 𝑓(⋅+𝑥), we have that 𝔼_𝑥𝑓𝑆_𝑎𝑏=𝔼𝑓𝑥+𝑆_{𝑎−𝑥,𝑏−𝑥}=^𝑁−1_𝑗=0𝐼⁻_{𝑎−𝑥,𝑏−𝑥,𝑗}(Δ⁻)^𝑗𝑓(𝑎)+^𝑁−1_𝑗=0𝐼⁺_{𝑎−𝑥,𝑏−𝑥,𝑗}Δ^+𝑗𝑓(𝑏). (332) More precisely, we have the following result.

Theorem 51. One has, for any function 𝑓 defined on ℰ, that 𝔼_𝑥𝑓𝑆_𝑎𝑏=^𝑁−1_𝑗=0𝑃⁻_{𝑎𝑏,𝑗}(𝑥)(Δ⁻)^𝑗𝑓(𝑎)+^𝑁−1_𝑗=0𝑃⁺_{𝑎𝑏,𝑗}(𝑥)Δ^+𝑗𝑓(𝑏), (333) where 𝑃⁻_{𝑎𝑏,𝑗} and 𝑃⁺_{𝑎𝑏,𝑗}, 0≤𝑗≤𝑁−1, are polynomials of degree not greater than 2𝑁−1 characterized, for any 𝑘∈{0,…,𝑁−1}, by (Δ⁻)^𝑘𝑃⁻_{𝑎𝑏,𝑗}(𝑎)=Δ^+𝑘𝑃⁺_{𝑎𝑏,𝑗}(𝑏)=𝛿_𝑗𝑘,(Δ⁻)^𝑘𝑃⁺_{𝑎𝑏,𝑗}(𝑎)=Δ^+𝑘𝑃⁻_{𝑎𝑏,𝑗}(𝑏)=0. (334)

Proof. By setting 𝑃⁻_{𝑎𝑏,𝑗}(𝑥)=𝐼⁻_{𝑎−𝑥,𝑏−𝑥,𝑗} and 𝑃⁺_{𝑎𝑏,𝑗}(𝑥)=𝐼⁺_{𝑎−𝑥,𝑏−𝑥,𝑗}, (330) immediately yields (333). By observing that 𝑃⁻_{𝑎𝑏,𝑗}(𝑥)=(−1)^𝑗𝑃⁺_{𝑎𝑏,𝑗}(𝑎+𝑏−𝑥), it is enough to work with, for example, 𝑃⁺_{𝑎𝑏,𝑗}. Coming back to the proof of Theorem 49 and appealing to Theorem 43, we write that 𝑃⁺_{𝑎𝑏,𝑗}(𝑥)=𝔼_𝑥𝟙_{{𝑆𝑎𝑏≥𝑏}}𝑆_𝑎𝑏𝑗=−𝑏_{𝑁−1𝑚=𝑗}𝑚𝑗ℙ_𝑥𝑆_𝑎𝑏=1=𝑏+𝑚()×𝑁−1!𝑁!_{𝑁−1𝑚=𝑗}(−1)^{𝑚𝑚𝑗𝑚𝑁−1}_{𝑁𝑚+𝑏−𝑎+𝑁−1}𝐾_𝑚(𝑥), (335) where, for any 𝑚∈{0,…,𝑁−1}, 𝐾_𝑚(𝑥)=∏^𝑁−1_𝑘=0[(𝑥−𝑎+𝑘)(𝑏−𝑥+𝑘)]()=𝑏−𝑥+𝑚_{𝑁−1𝑘=0}(𝑥−𝑎+𝑘)_{0≤𝑘≤𝑁−1𝑘≠𝑚}(𝑏−𝑥+𝑘). (336)
The expression 𝐾_𝑚(𝑥) defines a polynomial of the variable 𝑥 of degree 2𝑁−1, so 𝑃⁺_{𝑎𝑏,𝑗} is a polynomial of degree not greater than 2𝑁−1.
It is obvious that 𝐾_𝑚(𝑎−ℓ)=0 for ℓ,𝑚∈{0,…,𝑁−1}. Then, 𝑃⁺_{𝑎𝑏,𝑗}(𝑎−ℓ)=0 which implies that (Δ⁻)^𝑘𝑃⁺_{𝑎𝑏,𝑗}(𝑎)=0 for any 𝑘∈{0,…,𝑁−1}. Now, let us evaluate 𝑃⁺_{𝑎𝑏,𝑗}(𝑏+ℓ) for ℓ∈{0,…,𝑁−1}. We plainly have that 𝐾_𝑚(𝑏+ℓ)=0 for ℓ≠𝑚 and that 𝐾_ℓ()=()𝑏+ℓ−1^ℓ(𝑁−1)!𝑁!_𝑁^{ℓ+𝑏−𝑎+𝑁−1}_ℓ𝑁−1. (337) By putting this into (335), we get that 𝑃⁺_{𝑎𝑏,𝑗}()=ℓ𝑗𝑏+ℓ𝟙_{𝑗≤ℓ}. (338) Next, we obtain, for any 𝑘∈{0,…,𝑁−1}, that Δ^+𝑘𝑃⁺_{𝑎𝑏,𝑗}(𝑏)=^𝑘_ℓ=0(−1)^𝑘+ℓ𝑘ℓ𝑃⁺_{𝑎𝑏,𝑗}()=𝑏+ℓ_𝑘ℓ=𝑗(−1)^𝑘+ℓ𝑘ℓℓ𝑗=(−1)^𝑗+𝑘𝑘𝑗^{𝑘−𝑗ℓ=0}(−1)^ℓℓ𝑘−𝑗=𝛿_𝑗𝑘. (339) The proof of Theorem 51 is finished.

Example 52. In the case where 𝑁=2, (333) writes as 𝔼_𝑥𝑓𝑆_𝑎𝑏=𝑃⁻_𝑎𝑏,0(𝑥)𝑓(𝑎)+𝑃⁻_𝑎𝑏,1(𝑥)Δ⁻𝑓(𝑎)+𝑃⁺_𝑎𝑏,0(𝑥)𝑓(𝑏)+𝑃⁺_𝑎𝑏,1(𝑥)Δ⁺𝑓(𝑏) (340) with 𝑃⁻_𝑎𝑏,0(𝑥)=(𝑥−𝑏)(𝑥−𝑏−1)(2𝑥−3𝑎+𝑏+2)(),𝑃𝑏−𝑎)(𝑏−𝑎+1)(𝑏−𝑎+2_{−𝑎𝑏,1}(𝑥)=(𝑥−𝑎)(𝑥−𝑏)(𝑥−𝑏−1)(),𝑃𝑏−𝑎+1)(𝑏−𝑎+2_+𝑎𝑏,0(𝑥)()=−𝑥−𝑎)(𝑥−𝑎+1)(2𝑥+𝑎−3𝑏−2(),𝑃𝑏−𝑎)(𝑏−𝑎+1)(𝑏−𝑎+2_+𝑎𝑏,1(𝑥)=(𝑥−𝑎)(𝑥−𝑎+1)(𝑥−𝑏)().𝑏−𝑎+1)(𝑏−𝑎+2 (341)
Below, we state a strong pseudo-Markov property related to time 𝜎_𝑎𝑏.

Theorem 53. One has, for any function 𝑓 defined on ℤ and any 𝑛∈ℕ, that 𝔼_𝑥𝑓𝑆_{𝜎𝑎𝑏+𝑛}=^𝑁−1_𝑗=0𝑃⁻_{𝑎𝑏,𝑗}(𝑥)(Δ⁻)^𝑗𝔼_𝑎𝑓𝑆_𝑛+^𝑁−1_𝑗=0𝑃⁺_{𝑎𝑏,𝑗}(𝑥)Δ^+𝑗𝔼_𝑏𝑓𝑆_𝑛. (342) In (342), the operators (Δ⁻)^𝑗 and (Δ⁺)^𝑗 act on the variables 𝑎 and 𝑏.

Proof. Formula (342) can be proved exactly in the same way as (206): by setting 𝑔(𝑥)=𝔼_𝑥[𝑓(𝑆_𝑛)], we have that 𝔼_𝑥[𝑓(𝑆_{𝜎𝑎𝑏+𝑛})]=𝔼_𝑥[𝑔(𝑆_𝑎𝑏)]. This proves (342) thanks to (333).

Example 54. Below, we display the form of (342) for the particular values 1,2 of 𝑁. (i)For 𝑁=1, (206) reads as 𝔼_𝑥𝑓𝑆_{𝜎𝑎𝑏+𝑛}=𝑏−𝑥𝔼𝑏−𝑎^𝑎𝑓𝑆_𝑛+𝑥−𝑎𝔼𝑏−𝑎^𝑏𝑓𝑆_𝑛 (343) which is of course well known! This is the strong Markov property for the ordinary random walk.(ii)For 𝑁=2, (206) reads as 𝔼_𝑥𝑓𝑆_{𝜎𝑎𝑏+𝑛}=(𝑥−𝑏)(𝑥−𝑏−1)(2𝑥−3𝑎+𝑏+2)()𝔼𝑏−𝑎)(𝑏−𝑎+1)(𝑏−𝑎+2^𝑎𝑓𝑆_𝑛+(𝑥−𝑎)(𝑥−𝑏)(𝑥−𝑏−1)()Δ𝑏−𝑎+1)(𝑏−𝑎+2⁻𝔼_𝑎𝑓𝑆_𝑛−(𝑥−𝑎)(𝑥−𝑎+1)(2𝑥+𝑎−3𝑏−2)()𝔼𝑏−𝑎)(𝑏−𝑎+1)(𝑏−𝑎+2^𝑏𝑓𝑆_𝑛+(𝑥−𝑎)(𝑥−𝑎+1)(𝑥−𝑏)()Δ𝑏−𝑎+1)(𝑏−𝑎+2⁺𝔼_𝑏𝑓𝑆_𝑛=(𝑥−𝑎+1)(𝑥−𝑏)(𝑥−𝑏−1)()𝔼𝑏−𝑎)(𝑏−𝑎+1^𝑎𝑓𝑆_𝑛−(𝑥−𝑎)(𝑥−𝑏)(𝑥−𝑏−1)()𝔼𝑏−𝑎+1)(𝑏−𝑎+2^𝑎−1𝑓𝑆_𝑛−(𝑥−𝑎)(𝑥−𝑎+1)(𝑥−𝑏−1)()𝔼𝑏−𝑎)(𝑏−𝑎+1^𝑏𝑓𝑆_𝑛+(𝑥−𝑎)(𝑥−𝑎+1)(𝑥−𝑏)()𝔼𝑏−𝑎+1)(𝑏−𝑎+2^𝑏+1𝑓𝑆_𝑛. (344)
Now, we consider the discrete Laplacian Δ=Δ⁺∘Δ⁻=Δ⁻∘Δ⁺. It is explicitly defined by Δ𝑓(𝑖)=𝑓(𝑖+1)−2𝑓(𝑖)+𝑓(𝑖−1). Let us introduce the iterated Laplacian Δ^𝑁=(Δ⁺)^𝑁∘(Δ⁻)^𝑁=(Δ⁻)^𝑁∘(Δ⁺)^𝑁. We compute Δ^𝑁𝑓(𝑖) for any function 𝑓 and any 𝑖∈ℤ: Δ^𝑁𝑓(𝑖)=(Δ⁻)^𝑁𝑁_𝑗=0(−1)^𝑗+𝑁𝑁𝑗𝑓(𝑖)=⋅+𝑗_𝑁𝑗=0(−1)^𝑗+𝑁𝑁𝑗^𝑁𝑘=0(−1)^𝑘𝑁𝑘𝑓=()𝑖+𝑗−𝑘−1^𝑁_{0≤𝑗,𝑘≤𝑁}(−1)^𝑗−𝑘𝑁𝑗𝑁𝑘𝑓=()𝑖+𝑗−𝑘−1^𝑁𝑁_ℓ=−𝑁(−1)^{ℓ(𝑁−ℓ)∧𝑁}_{()𝑘=−ℓ∨0}𝑁𝑘𝑁𝑓().𝑘+ℓ𝑖+ℓ (345) By using the elementary identity ∑^𝑝∧ℓ_{𝑘=(ℓ−𝑞)∨0}^𝑝_𝑘^𝑞_ℓ−𝑘=_ℓ^𝑝+𝑞, we get that ^{(𝑁−ℓ)∧𝑁}_{()𝑘=−ℓ∨0}𝑁𝑘𝑁=𝑘+ℓ^{(𝑁−ℓ)∧𝑁()𝑘=−ℓ∨0}𝑁𝑘𝑁=.𝑁−𝑘−ℓ2𝑁ℓ+𝑁 (346) As a result, we obtain the expression of Δ^𝑁𝑓(𝑖) announced in the introduction, namely, Δ^𝑁𝑓(𝑖)=^𝑁_ℓ=−𝑁(−1)^ℓ+𝑁𝑓().2𝑁ℓ+𝑁𝑖+ℓ (347)

Example 55. Fix a nonnegative integer 𝑗 and put 𝑓_𝑗(𝑖)=(𝑖)_𝑗 for any 𝑖∈ℤ. It is plain that, if 𝑘≤𝑗, (Δ⁺)^𝑘𝑓_𝑗(𝑖)=(𝑗)_𝑘𝑓_𝑗−𝑘(𝑖),(Δ⁻)^𝑘𝑓_𝑗(𝑖)=(𝑗)_𝑘𝑓_𝑗−𝑘(𝑖−1) and if 𝑘>𝑗, (Δ⁺)^𝑘𝑓_𝑗(𝑖)=(Δ⁻)^𝑘𝑓_𝑗(𝑖)=0. Therefore, if 2𝑘≤𝑗, Δ^𝑘𝑓_𝑗(𝑖)=(𝑗)_𝑘(𝑗−𝑘)_𝑘𝑓_𝑗−2𝑘(𝑖−1) and if 2𝑘>𝑗, Δ^𝑘𝑓_𝑗(𝑖)=0. By using a linear algebra argument, we deduce that Δ^𝑁𝑃=0 for any polynomial 𝑃 of degree not greater than 2𝑁−1. As a byproduct, Δ^𝑁𝑃⁺_{𝑎𝑏,𝑗}=Δ^𝑁𝑃⁻_{𝑎𝑏,𝑗}=0. (348)

Now, the main link between time 𝜎_𝑎𝑏 and finite-difference equations is the following one.

Theorem 56. Let 𝜑 be a function defined on ℰ. The function Φ defined on ℤ by Φ(𝑥)=𝔼_𝑥[𝜑(𝑆_𝑎𝑏)] is the solution to the discrete Lauricella’s problem Δ^𝑁Φ(𝑥)(Δ=0𝑓𝑜𝑟𝑥∈ℤ,₋)^𝑘Φ(𝑎)=(Δ⁻)^𝑘𝜑(𝑎){},Δ𝑓𝑜𝑟𝑗∈0,…,𝑁−1_+𝑘Φ(𝑏)=Δ^+𝑘𝜑(𝑏)𝑓𝑜𝑟𝑗∈{0,…,𝑁−1}. (349)

Proof. By (333) we write that Φ(𝑥)=^𝑁−1_𝑗=0𝑃⁻_{𝑎𝑏,𝑗}(𝑥)(Δ⁻)^𝑗𝜑(𝑎)+^𝑁−1_𝑗=0𝑃⁺_{𝑎𝑏,𝑗}(𝑥)Δ^+𝑗𝜑(𝑏). (350) With this representation at hand, identities (334) and (348) immediately yield equations (349).

4.3. Joint Pseudodistribution of (𝜏_𝑎𝑏,𝑋_𝑎𝑏)

As in Section 3.3, we choose for the family ((𝑋^𝜀_𝑡)_𝑡≥0)_𝜀>0 the pseudoprocesses defined, for any 𝜀>0, by 𝑋^𝜀_𝑡=𝜀𝑆_{⌊𝑡/𝜀2𝑁⌋},𝑡≥0, (351)

and for the pseudoprocess (𝑋_𝑡)_𝑡≥0 the pseudo-Brownian motion. In Definition 34, we choose for 𝐼 the interval (𝑎,𝑏); then 𝜏^𝜀_𝐼=𝜏^𝜀_𝑎𝑏, 𝑋^𝜀_𝐼=𝑋^𝜀_𝑎𝑏 and 𝜏_𝐼=𝜏_𝑎𝑏, 𝑋_𝐼=𝑋_𝑎𝑏. Set 𝑎_𝜀=⌊𝑎/𝜀⌋ and 𝑏_𝜀=⌊𝑏/𝜀⌋, where ⌊⋅⌋ and ⌈⋅⌉, respectively stand for the usual floor and ceiling functions. We have 𝜏^𝜀_𝑎𝑏=𝜀^2𝑁𝜎_{𝑎𝜀,𝑏𝜀} and 𝑋^𝜀_𝑎𝑏=𝜀𝑆_{𝑎𝜀,𝑏𝜀}.

Theorem 57. The following convergence holds: 𝜏^𝜀_𝑎𝑏,𝑋^𝜀_𝑎𝑏⟶_𝜀→0+𝜏_𝑎𝑏,𝑋_𝑎𝑏, (352) where, for any 𝜆>0 and any 𝜇∈ℝ, 𝔼e^{−𝜆𝜏𝑎𝑏+𝑖𝜇𝑋𝑎𝑏}𝟙_{{𝜏𝑎𝑏<+∞}}=e^{𝑁𝑖𝜇𝑎}_𝑘=1𝐃⁻_𝑘𝜆,𝜇𝐃(𝜆)+e^{𝑁𝑖𝜇𝑏}_𝑘=1𝐃⁺_𝑘𝜆,𝜇𝐃(𝜆). (353) In the foregoing formula, 𝐃(𝜆), 𝐃⁻_𝑘(𝜆,𝜇), and 𝐃⁺_𝑘(𝜆,𝜇) are the respective determinants1𝜑₁⋯𝜑₁^𝑁−1e^{−𝜑12𝑁𝜆/𝑐(𝑏−𝑎)}e^{−𝜑12𝑁𝜆/𝑐(𝑏−𝑎)}𝜑₁⋯e^{−𝜑12𝑁𝜆/𝑐(𝑏−𝑎)}𝜑₁^𝑁−1⋮⋮⋮⋮⋮⋮1𝜑_2𝑁⋯𝜑^𝑁−1_2𝑁e^{−𝜑2𝑁2𝑁𝜆/𝑐(𝑏−𝑎)}e^{−𝜑2𝑁2𝑁𝜆/𝑐(𝑏−𝑎)}𝜑_2𝑁⋯e^{−𝜑2𝑁2𝑁𝜆/𝑐(𝑏−𝑎)}𝜑^𝑁−1_2𝑁,1𝜑₁⋯𝜑₁^𝑁−1e^{−𝜑12𝑁𝜆/𝑐(𝑏−𝑎)}e^{−𝜑12𝑁𝜆/𝑐(𝑏−𝑎)}𝜑₁⋯e^{−𝜑12𝑁𝜆/𝑐(𝑏−𝑎)}𝜑₁^𝑁−1⋮⋮⋮⋮⋮⋮1𝜑_𝑘−1⋯𝜑^𝑁−1_𝑘−1e^{−𝜑𝑘−12𝑁𝜆/𝑐(𝑏−𝑎)}e^{−𝜑𝑘−12𝑁𝜆/𝑐(𝑏−𝑎)}𝜑_𝑘−1⋯e^{−𝜑𝑘−12𝑁𝜆/𝑐(𝑏−𝑎)}𝜑^𝑁−1_𝑘−11𝛿⋯𝛿^𝑁−100⋯01𝜑_𝑘+1⋯𝜑^𝑁−1_𝑘+1e^{−𝜑𝑘+12𝑁𝜆/𝑐(𝑏−𝑎)}e^{−𝜑𝑘+12𝑁𝜆/𝑐(𝑏−𝑎)}𝜑_𝑘+1⋯e^{−𝜑𝑘+12𝑁𝜆/𝑐(𝑏−𝑎)}𝜑^𝑁−1_𝑘+1⋮⋮⋮⋮⋮⋮1𝜑_2𝑁⋯𝜑^𝑁−1_2𝑁e^{−𝜑2𝑁2𝑁𝜆/𝑐(𝑏−𝑎)}e^{−𝜑2𝑁2𝑁𝜆/𝑐(𝑏−𝑎)}𝜑_2𝑁⋯e^{−𝜑2𝑁2𝑁𝜆/𝑐(𝑏−𝑎)}𝜑^𝑁−1_2𝑁,1𝜑₁⋯𝜑₁^𝑁−1e^{−𝜑12𝑁𝜆/𝑐(𝑏−𝑎)}e^{−𝜑12𝑁𝜆/𝑐(𝑏−𝑎)}𝜑₁⋯e^{−𝜑12𝑁𝜆/𝑐(𝑏−𝑎)}𝜑₁^𝑁−1⋮⋮⋮⋮⋮⋮1𝜑_𝑘−1⋯𝜑^𝑁−1_𝑘−1e^{−𝜑𝑘−12𝑁𝜆/𝑐(𝑏−𝑎)}e^{−𝜑𝑘−12𝑁𝜆/𝑐(𝑏−𝑎)}𝜑_𝑘−1⋯e^{−𝜑𝑘−12𝑁𝜆/𝑐(𝑏−𝑎)}𝜑^𝑁−1_𝑘−100⋯01𝛿⋯𝛿^𝑁−11𝜑_𝑘+1⋯𝜑^𝑁−1_𝑘+1e^{−𝜑𝑘+12𝑁𝜆/𝑐(𝑏−𝑎)}e^{−𝜑𝑘+12𝑁𝜆/𝑐(𝑏−𝑎)}𝜑_𝑘+1⋯e^{−𝜑𝑘+12𝑁𝜆/𝑐(𝑏−𝑎)}𝜑^𝑁−1_𝑘+1⋮⋮⋮⋮⋮⋮1𝜑_2𝑁⋯𝜑^𝑁−1_2𝑁e^{−𝜑2𝑁2𝑁𝜆/𝑐(𝑏−𝑎)}e^{−𝜑2𝑁2𝑁𝜆/𝑐(𝑏−𝑎)}𝜑_2𝑁⋯e^{−𝜑2𝑁2𝑁𝜆/𝑐(𝑏−𝑎)}𝜑^𝑁−1_2𝑁. (354)
In the two last determinants, we have put 𝛿=−𝑖𝜇/^2𝑁𝜆/𝑐.

In Theorem 57, we obtain the joint pseudodistribution of (𝜏_𝑎𝑏,𝑋_𝑎𝑏) characterized by its Laplace-Fourier transform. This is a new result for pseudo-Brownian motion that we will develop in a forthcoming paper [14].

Proof. By Definition 34 and Theorem 39, we have that 𝔼e^{−𝜆𝜏𝑎𝑏+𝑖𝜇𝑋𝑎𝑏}𝟙_{{𝜏𝑎𝑏<+∞}}=lim_𝜀→0+𝔼e^{−𝜆𝜏𝜀𝑎𝑏+𝑖𝜇𝑋𝜀𝑎𝑏}𝟙_{𝜏^𝜀_{𝑎𝑏<+∞}}=lim_𝜀→0+𝔼e^{−𝜆𝜀2𝑁𝜎𝑎𝜀𝜀,𝑏+𝑖𝜇𝜀𝑆𝑎𝜀𝜀,𝑏}𝟙_{{𝜎𝑎𝜀𝜀,𝑏<+∞}}=lim_𝜀→0+^2𝑁_𝑘=1𝐿_𝑘e^{−𝜆𝜀2𝑁},e^𝑖𝜇𝜀e^𝑖𝜇𝜀𝑣_𝑘(𝜆,𝜀)^{𝑎𝜀−𝑁}=e^{𝑖𝜇𝑎2𝑁}_𝑘=1lim_𝜀→0+𝐿_𝑘e^{−𝜆𝜀2𝑁},e^𝑖𝜇𝜀𝑣_𝑘(𝜆,𝜀)^𝑎𝜀. (355)
Recall that 𝐿_𝑘(𝑧,𝜁)=𝐷_𝑘(𝑧,𝜁)/𝐷(𝑧) and that the quantities 𝐷 and 𝐷_𝑘 are expressed by means of the determinant𝑈𝑢₁,…,𝑢_2𝑁=1𝑢₁𝑢²₁⋯𝑢₁^𝑁−1𝑢₁^{𝑏−𝑎+𝑁−1}𝑢₁^{𝑏−𝑎+𝑁}𝑢₁^{𝑏−𝑎+𝑁+1}⋯𝑢₁^{𝑏−𝑎+2𝑁−2}⋮⋮⋮⋮⋮⋮⋮⋮1𝑢_2𝑁𝑢²_2𝑁⋯𝑢^𝑁−1_2𝑁𝑢^{𝑏−𝑎+𝑁−1}_2𝑁𝑢^{𝑏−𝑎+𝑁}_2𝑁𝑢^{𝑏−𝑎+𝑁+1}_2𝑁⋯𝑢^{𝑏−𝑎+2𝑁−2}_2𝑁. (356)
By replacing the columns labeled as 𝐶_𝑗, 1≤𝑗≤2𝑁, by the linear combinations 𝐶^′_𝑗=∑^𝑗_𝑘=0(−1)^{𝑗+𝑘𝑗}_𝑘𝐶_𝑘 if 1≤𝑗≤𝑁, and 𝐶^′_𝑗=∑^𝑗_𝑘=𝑁(−1)^{𝑗+𝑘𝑗}_𝑘𝐶_𝑘 if 𝑁+1≤𝑗≤2𝑁, the foregoing determinant remains invariant and can be rewritten as^{1𝑢1𝑢−11−12⋯𝑢1−1𝑁−1𝑢1𝑏−𝑎+𝑁−1𝑢1𝑏−𝑎+𝑁−1()𝑢𝑢−11𝑏−𝑎+𝑁−1()𝑢−12⋯𝑢1𝑏−𝑎+2𝑁−2𝑢1−1𝑁−1}_{⋮⋮⋮⋮⋮⋮⋮⋮1𝑢2𝑁𝑢−12𝑁−12⋯𝑢2𝑁−1𝑁−1𝑢𝑏−𝑎+𝑁−12𝑁𝑢𝑏−𝑎+𝑁−12𝑁()𝑢𝑢−1𝑏−𝑎+𝑁−12𝑁()𝑢−12⋯𝑢𝑏−𝑎+2𝑁−22𝑁𝑢1−1𝑁−1}. (357)
Then, by replacing the 𝑢_𝑗’s by 𝑢_𝑗(𝜆,𝜀), 𝑏−𝑎 by 𝑏_𝜀−𝑎_𝜀 and by using the asymptotics (𝑢_𝑗(𝜆,𝜀)−1)^𝑘∼_𝜀→0+(−𝜑_𝑗^2𝑁𝜆/𝑐)^𝑘𝜀^𝑘 and 𝑢_𝑗(𝜆,𝜀)^{𝑏𝜀−𝑎𝜀+𝑁−1}→_𝜀→0+e^{−𝜑𝑗2𝑁𝜆/𝑐(𝑏−𝑎)} coming from (119), we get that𝐷e^{−𝜆𝜀2𝑁}=𝑈𝑢₁(𝜆,𝜀),…,𝑢_2𝑁()∼𝜆,𝜀_𝜀→0+1−𝜑₁^2𝑁𝜆𝑐𝜀⋯−𝜑₁^2𝑁𝜆𝑐^𝑁−1𝜀^𝑁−1e^{−𝜑12𝑁𝜆/𝑐(𝑏−𝑎)}1⋮⋮⋮⋮−𝜑_2𝑁^2𝑁𝜆𝑐𝜀⋯−𝜑_2𝑁^2𝑁𝜆𝑐^𝑁−1𝜀^𝑁−1e^{−𝜑2𝑁2𝑁𝜆/𝑐(𝑏−𝑎)}−𝜑₁^2𝑁𝜆𝑐e^{−𝜑12𝑁𝜆/𝑐(𝑏−𝑎)}𝜀⋯−𝜑₁^2𝑁𝜆𝑐^𝑁−1e^{−𝜑12𝑁𝜆/𝑐(𝑏−𝑎)}𝜀^𝑁−1⋮⋮−𝜑_2𝑁^2𝑁𝜆𝑐e^{−𝜑2𝑁2𝑁𝜆/𝑐(𝑏−𝑎)}𝜀⋯−𝜑_2𝑁^2𝑁𝜆𝑐^𝑁−1e^{−𝜑2𝑁2𝑁𝜆/𝑐(𝑏−𝑎)}𝜀^𝑁−1=𝜆𝑐^(𝑁−1)/2𝜀^{𝑁(𝑁−1)}𝐃(𝜆). (358)
Similarly, by using the elementary asymptotics e^𝑖𝜇𝜀−1∼_𝜀→0+𝑖𝜇𝜀 and e^{𝑖𝜇𝜀(𝑏𝜀−𝑎𝜀+𝑁−1)}→_𝜀→0+e^{𝑖𝜇(𝑏−𝑎)}, we obtain that e^𝑖𝜇𝑎𝐷_𝑘e^{−𝜆𝜀2𝑁},e^𝑖𝜇𝜀=e^𝑖𝜇𝑎𝑈𝑢₁(𝜆,𝜀),…,𝑢_𝑘−1(𝜆,𝜀),e^𝑖𝜇𝜀,𝑢_𝑘+1(𝜆,𝜀),…,𝑢_2𝑁()∼𝜆,𝜀_𝜀→0+𝜆𝑐^(𝑁−1)/2𝜀^{𝑁(𝑁−1)}𝐃_𝑘𝜆,𝜇, (359)
where 𝐃_𝑘(𝜆,𝜇) denotes the determinant1𝜑₁⋯𝜑₁^𝑁−1e^{−𝜑12𝑁𝜆/𝑐(𝑏−𝑎)}e^{−𝜑12𝑁𝜆/𝑐(𝑏−𝑎)}𝜑₁⋯e^{−𝜑12𝑁𝜆/𝑐(𝑏−𝑎)}𝜑₁^𝑁−1⋮⋮⋮⋮⋮⋮1𝜑_𝑘−1⋯𝜑^𝑁−1_𝑘−1e^{−𝜑𝑘−12𝑁𝜆/𝑐(𝑏−𝑎)}e^{−𝜑𝑘−12𝑁𝜆/𝑐(𝑏−𝑎)}𝜑_𝑘−1⋯e^{−𝜑𝑘−12𝑁𝜆/𝑐(𝑏−𝑎)}𝜑^𝑁−1_𝑘−1e^𝑖𝜇𝑎e^𝑖𝜇𝑎𝛿⋯e^𝑖𝜇𝑎𝛿^𝑁−1e^𝑖𝜇𝑏e^𝑖𝜇𝑏𝛿⋯e^𝑖𝜇𝑏𝛿^𝑁−11𝜑_𝑘+1⋯𝜑^𝑁−1_𝑘+1e^{−𝜑𝑘+12𝑁𝜆/𝑐(𝑏−𝑎)}e^{−𝜑𝑘+12𝑁𝜆/𝑐(𝑏−𝑎)}𝜑_𝑘+1⋯e^{−𝜑𝑘+12𝑁𝜆/𝑐(𝑏−𝑎)}𝜑^𝑁−1_𝑘+1⋮⋮⋮⋮⋮⋮1𝜑_2𝑁⋯𝜑^𝑁−1_2𝑁e^{−𝜑2𝑁2𝑁𝜆/𝑐(𝑏−𝑎)}e^{−𝜑2𝑁2𝑁𝜆/𝑐(𝑏−𝑎)}𝜑_2𝑁⋯e^{−𝜑2𝑁2𝑁𝜆/𝑐(𝑏−𝑎)}𝜑^𝑁−1_2𝑁. (360)
By putting (358) and (359) into (355), we derive that 𝔼e^{−𝜆𝜏𝑎𝑏+𝑖𝜇𝑋𝑎𝑏}𝟙_{{𝜏𝑎𝑏<+∞}}=^2𝑁_𝑘=1𝐃_𝑘𝜆,𝜇𝐃(𝜆). (361)
It is plain that 𝐃_𝑘(𝜆,𝜇)=e^𝑖𝜇𝑎𝐃⁻_𝑘(𝜆,𝜇)+e^𝑖𝜇𝑏𝐃⁻_𝑘(𝜆,𝜇) which finishes the proof of Theorem 57.

Theorem 58. The following convergence holds: 𝑋^𝜀_𝑎𝑏⟶_𝜀→0+𝑋_𝑎𝑏, (362) where, for any 𝜇∈ℝ, 𝔼e^{𝑖𝜇𝑋𝑎𝑏}𝟙_{{𝜏𝑎𝑏<+∞}}=e^{𝑖𝜇𝑎𝑁−1}_𝑗=0𝐈⁻_{𝑎𝑏,𝑗}𝑖𝜇^𝑗+e^{𝑖𝜇𝑏𝑁−1}_𝑗=0𝐈⁺_{𝑎𝑏,𝑗}𝑖𝜇^𝑗 (363) with 𝐈⁻_{𝑎𝑏,𝑗}=𝑏𝑏−𝑎^𝑁(−𝑎)^𝑗𝑗!^{𝑁−𝑗−1}_𝑘=0𝑘𝑘+𝑁−1−𝑎𝑏−𝑎^𝑘,𝐈⁺_{𝑎𝑏,𝑗}=−𝑎𝑏−𝑎^𝑁(−𝑏)^𝑗𝑗!^{𝑁−𝑗−1}_𝑘=0𝑘𝑘+𝑁−1𝑏𝑏−𝑎^𝑘. (364) Moreover, ℙ𝜏_𝑎𝑏<+∞=1. (365)

Proof. By Definition 34 and by (331), 𝔼e^{𝑖𝜇𝑋𝑎𝑏}𝟙_{{𝜏𝑎𝑏<+∞}}=lim_𝜀→0+𝔼e^{𝑖𝜇𝑋𝜀𝑎𝑏}𝟙_{𝜏^𝜀_{𝑎𝑏<+∞}}=lim_𝜀→0+𝔼e^{𝑖𝜇𝜀𝑆𝑎𝜀𝜀,𝑏}𝟙_{{𝜎𝑎𝜀𝜀,𝑏<+∞}}=lim_𝜀→0+e^{𝑖𝜇𝜀𝑎𝜀𝑁−1}_𝑗=0𝐼⁻_{𝑎𝜀,𝑏𝜀,𝑗}1−e^{−𝑖𝜇𝜀𝑗}+e^{𝑖𝜇𝜀𝑏𝜀𝑁−1}_𝑗=0𝐼⁺_{𝑎𝜀,𝑏𝜀,𝑗}e^𝑖𝜇𝜀−1^𝑗=e^𝑖𝜇𝑎lim_𝜀→0+^𝑁−1_𝑗=0𝐼⁻_{𝑎𝜀,𝑏𝜀,𝑗}1−e^{−𝑖𝜇𝜀𝑗}+e^𝑖𝜇𝑏lim_𝜀→0+^𝑁−1_𝑗=0𝐼⁺_{𝑎𝜀,𝑏𝜀,𝑗}e^𝑖𝜇𝜀−1^𝑗. (366) Concerning, for example, the quantity 𝐼⁺_{𝑎𝜀,𝑏𝜀,𝑗}, we have that 𝐼⁺_{𝑎𝜀,𝑏𝜀,𝑗}=(−1)^𝑗(𝑁−1)!²𝑗𝑁−1^𝑁−1_𝑘=0𝑘−𝑎_𝜀𝑘+𝑏_𝜀×_𝒟⁺𝑢^{−𝑎𝜀−1}(1−𝑢)^𝑁−1𝑣^{𝑗+𝑏𝜀−1}(1−𝑣)^{𝑁−𝑗−1}d𝑢d𝑣. (367) By performing the change of variables (𝑢,𝑣)=(𝑥,𝑥𝑦) in the above integral and by expanding (1−𝑥𝑦)^{𝑁−𝑗−1} as 1−𝑥𝑦^{𝑁−𝑗−1}=(1−𝑥)+𝑥1−𝑦^{𝑁−𝑗−1}=^{𝑁−𝑗−1}_𝑘=0𝑘()𝑁−𝑗−11−𝑥^𝑘𝑥^{𝑁−𝑗−𝑘−1}1−𝑦^{𝑁−𝑗−𝑘−1}, (368) we get that _𝒟⁺𝑢^{−𝑎𝜀−1}(1−𝑢)^𝑁−1𝑣^{𝑗+𝑏𝜀−1}(1−𝑣)^{𝑁−𝑗−1}d𝑢d𝑣=¹₀¹₀𝑥^{𝑗+𝑏𝜀−𝑎𝜀−1}(1−𝑥)^𝑁−1𝑦^{𝑗+𝑏𝜀−1}×1−𝑥𝑦^{𝑁−𝑗−1}d𝑥d𝑦=^{𝑁−𝑗−1}_𝑘=0𝑘×𝑁−𝑗−1₁₀𝑥^{𝑏𝜀−𝑎𝜀+𝑁−𝑘−2}(1−𝑥)^{𝑘+𝑁−1}d𝑥×¹₀𝑦^{𝑗+𝑏𝜀−1}1−𝑦^{𝑁−𝑗−𝑘−1}d𝑦=^{𝑁−𝑗−1}_𝑘=0𝑘×𝑏𝑁−𝑗−1_𝜀−𝑎_𝜀+𝑁−𝑘−2!(𝑘+𝑁−1)!𝑏_𝜀−𝑎_𝜀!×+2𝑁−2𝑗+𝑏_𝜀−1!𝑁−𝑗−𝑘−1!𝑏_𝜀!=()!!+𝑁−𝑘−1𝑁−1𝑁−𝑗−1𝑗+𝑏_𝜀−1!𝑏_𝜀−𝑎_𝜀!×+2𝑁−2_{𝑁−𝑗−1𝑘=0}𝑘𝑘+𝑁−1𝑏_𝜀−𝑎_𝜀+𝑁−𝑘−2!𝑏_𝜀!.+𝑁−𝑘−1 (369) By putting the asymptotics 𝑗+𝑏_𝜀−1!𝑏_𝜀!=1+𝑁−𝑘−1𝑏_𝜀+𝑁−𝑘−1_{𝑁−𝑗−𝑘𝜀→}∼₀₊𝜀𝑏^{𝑁−𝑗−𝑘},𝑏_𝜀−𝑎_𝜀+𝑁−𝑘−2!𝑏_𝜀−𝑎_𝜀!=1+2𝑁−2𝑏_𝜀−𝑎_𝜀+2𝑁−2_{𝑁+𝑘𝜀→}∼₀₊𝜀𝑏−𝑎^𝑁+𝑘 (370) into (369), we obtain that _𝒟⁺𝑢^{−𝑎𝜀−1}(1−𝑢)^𝑁−1𝑣^{𝑗+𝑏𝜀−1}(1−𝑣)^{𝑁−𝑗−1}d𝑢d𝑣_𝜀→∼₀₊(𝑁−1)!𝑁−𝑗−1!𝑏^𝑁−𝑗(𝑏−𝑎)^𝑁×𝜀^{2𝑁−𝑗𝑁−𝑗−1}_𝑘=0𝑘𝑘+𝑁−1𝑏𝑏−𝑎^𝑘. (371) Next, using the asymptotics ^𝑁−1_𝑘=0𝑘−𝑎_𝜀𝑘+𝑏_𝜀∼_𝜀→0+(−1)^𝑁(𝑎𝑏)^𝑁𝜀^2𝑁, (372) expression (367) admits the following asymptotics: 𝐼⁺_{𝑎𝜀,𝑏𝜀,𝑗}∼_𝜀→0+(−1)^𝑗+𝑁𝑏^𝑗𝑗!𝜀^𝑗𝑎𝑏−𝑎^𝑁×^{𝑁−𝑗−1}_𝑘=0𝑘𝑘+𝑁−1𝑏𝑏−𝑎^𝑘. (373) Then, we see that the second limit lying in (366) equals −𝑎𝑏−𝑎^𝑁𝑁−1_𝑗=0−𝑖𝜇𝑏^𝑗𝑗!^{𝑁−𝑗−1}_𝑘=0𝑘𝑘+𝑁−1𝑏𝑏−𝑎^𝑘. (374) In the same way, it may be seen that the first term of the sum lying in (366) tends to 𝑏𝑏−𝑎^𝑁𝑁−1_𝑗=0−𝑖𝜇𝑎^𝑗𝑗!^{𝑁−𝑗−1}_𝑘=0𝑘𝑘+𝑁−1−𝑎𝑏−𝑎^𝑘. (375)
As a result, we derive (363).
Finally, let us have a look on the pseudoprobability ℙ{𝜏_𝑎𝑏<+∞}. We have that 𝐈⁻_𝑎𝑏,0=𝑏𝑏−𝑎^𝑁𝑁−1_𝑘=0𝑘𝑘+𝑁−1−𝑎𝑏−𝑎^𝑘=𝑏^𝑁(𝑏−𝑎)^{2𝑁−1𝑁−1}_𝑘=0𝑘)𝑘+𝑁−1(−𝑎^𝑘(𝑏−𝑎)^{𝑁−1−𝑘}=𝑏^𝑁(𝑏−𝑎)^2𝑁−1_{0≤𝑘≤𝑁−10≤ℓ≤𝑁−1−𝑘}𝑘ℓ×()𝑘+𝑁−1𝑁−1−𝑘−𝑎^𝑘+ℓ𝑏^{𝑁−1−𝑘−ℓ}=1(𝑏−𝑎)^{2𝑁−1𝑁−1}_𝑚=0^𝑚_𝑘=0𝑘×()𝑘+𝑁−1𝑁−1−𝑘𝑚−𝑘−𝑎^𝑚𝑏^{2𝑁−1−𝑚}. (376) By using the elementary identity ∑^𝑛_𝑘=0𝑘^{𝑘+𝑝𝑛+𝑞−𝑘}_𝑛−𝑘=_𝑛^{𝑛+𝑝+𝑞+1} which comes from the equality (1+𝑥)^−𝑝(1+𝑥)^−𝑞=(1+𝑥)^{−𝑝−𝑞} together with the expansion, for example, for 𝑝, (1+𝑥)^−𝑝=∑^∞_𝑘=0(−1)^𝑘_𝑘^{𝑘+𝑝−1}𝑥^𝑘, we get that ^𝑚𝑘=0𝑘=𝑚.𝑘+𝑁−1𝑁−1−𝑘𝑚−𝑘2𝑁−1 (377) As a byproduct, 𝐈⁻_𝑎𝑏,0=1(𝑏−𝑎)^{2𝑁−1𝑁−1}_𝑚=0𝑚()2𝑁−1−𝑎^𝑚𝑏^{2𝑁−1−𝑚}. (378) Similarly, 𝐈⁺_𝑎𝑏,0=1(𝑏−𝑎)^{2𝑁−12𝑁−1}_𝑚=𝑁𝑚()2𝑁−1−𝑎^𝑚𝑏^{2𝑁−1−𝑚} (379) and we deduce that ℙ𝜏_𝑎𝑏<+∞=𝐈⁻_𝑎𝑏,0+𝐈⁺_𝑎𝑏,0=1(𝑏−𝑎)^{2𝑁−12𝑁−1}_𝑚=0𝑚)2𝑁−1(−𝑎^𝑚𝑏^{2𝑁−1−𝑚}=1. (380) The proof of Theorem 58 is finished.

Corollary 59. The pseudodensity of 𝑋_𝑎𝑏 is given by ℙ𝑋_𝑎𝑏∈d𝑧d𝑧=^𝑁−1_𝑗=0(−1)^𝑗𝐈⁻_{𝑎𝑏,𝑗}𝛿_𝑎^(𝑗)(𝑧)+^𝑁−1_𝑗=0(−1)^𝑗𝐈⁺_{𝑎𝑏,𝑗}𝛿_𝑏^(𝑗)(𝑧). (381) In particular, ℙ𝜏⁻_𝑎<𝜏⁺_𝑏=𝐈⁻_𝑎𝑏,0,ℙ𝜏⁺_𝑏<𝜏⁻_𝑎=𝐈⁺_𝑎𝑏,0. (382)

This result has been announced in [13] without any proof. We will develop it in a forthcoming paper.

Appendix

A.

A.1. Lacunary Vandermonde Systems

Let us introduce the “lacunary” Vandermonde determinant (of type (𝑝+𝑟)×(𝑝+𝑟)): 𝑈_𝑝𝑞𝑟𝑢₁,…,𝑢_𝑝+𝑟=1𝑢₁⋯𝑢₁^𝑝−1𝑢₁^𝑝+𝑞⋯𝑢₁^{𝑝+𝑞+𝑟−1}⋮⋮⋮⋮⋮1𝑢_𝑝+𝑟⋯𝑢^𝑝−1_𝑝+𝑟𝑢^𝑝+𝑞_𝑝+𝑟⋯𝑢^{𝑝+𝑞+𝑟−1}_𝑝+𝑟. (A.1) We put 𝑠₀∶=𝑠₀(𝑢₁,…,𝑢_𝑝+𝑟)=1 and, for ℓ∈{1,…,𝑝+𝑟}, 𝑠_ℓ∶=𝑠_ℓ𝑢₁,…,𝑢_𝑝+𝑟=_{1≤𝑖1<⋯<𝑖ℓ≤𝑝+𝑟}𝑢_𝑖1⋯𝑢_𝑖ℓ. (A.2) We say “lacunary” because it comes from a genuine Vandermonde determinant where the powers from 𝑝 to (𝑝+𝑞−1) are missing. More precisely, the determinant 𝑈_𝑝𝑞𝑟(𝑢₁,…,𝑢_𝑝+𝑟) is extracted from the classical Vandermonde determinant 𝑉_{𝑝+𝑞+𝑟}(𝑢₁,…,𝑢_{𝑝+𝑞+𝑟}) of type (𝑝+𝑞+𝑟)×(𝑝+𝑞+𝑟) by removing the 𝑞 last rows and the (𝑝+1)st, (𝑝+2)nd,…,(𝑝+𝑞)th columns. We decompose 𝑉_{𝑝+𝑞+𝑟}(𝑢₁,…,𝑢_{𝑝+𝑞+𝑟}) into blocks as follows:1𝑢₁⋯𝑢₁^𝑝−1𝑢^𝑝₁⋯𝑢₁^{𝑝+𝑞−1}𝑢₁^𝑝+𝑞⋯𝑢₁^{𝑝+𝑞+𝑟−1}⋮⋮⋮⋮⋮⋮⋮1𝑢_𝑝+𝑟⋯𝑢^𝑝−1_𝑝+𝑟𝑢^𝑝_𝑝+𝑟⋯𝑢^{𝑝+𝑞−1}_𝑝+𝑟𝑢^𝑝+𝑞_𝑝+𝑟⋯𝑢^{𝑝+𝑞+𝑟−1}_𝑝+𝑟1𝑢_𝑝+𝑟+1⋯𝑢^𝑝−1_𝑝+𝑟+1𝑢^𝑝_𝑝+𝑟+1⋯𝑢^{𝑝+𝑞−1}_𝑝+𝑟+1𝑢^𝑝+𝑞_𝑝+𝑟+1⋯𝑢^{𝑝+𝑞+𝑟−1}_𝑝+𝑟+1⋮⋮⋮⋮⋮⋮⋮1𝑢_{𝑝+𝑞+𝑟}⋯𝑢^𝑝−1_{𝑝+𝑞+𝑟}𝑢^𝑝_{𝑝+𝑞+𝑟}⋯𝑢^{𝑝+𝑞−1}_{𝑝+𝑞+𝑟}𝑢^𝑝+𝑞_{𝑝+𝑞+𝑟}⋯𝑢^{𝑝+𝑞+𝑟−1}_{𝑝+𝑞+𝑟}. (A.3)

By moving the 𝑟 last columns before the 𝑞 previous ones, this determinant can be rewritten as(−1)^𝑞𝑟1𝑢₁⋯𝑢₁^𝑝−1𝑢₁^𝑝+𝑞⋯𝑢₁^{𝑝+𝑞+𝑟−1}𝑢^𝑝₁⋯𝑢₁^{𝑝+𝑞−1}⋮⋮⋮⋮⋮⋮⋮1𝑢_𝑝+𝑟⋯𝑢^𝑝−1_𝑝+𝑟𝑢^𝑝+𝑞_𝑝+𝑟⋯𝑢^{𝑝+𝑞+𝑟−1}_𝑝+𝑟𝑢^𝑝_𝑝+𝑟⋯𝑢^{𝑝+𝑞−1}_𝑝+𝑟1𝑢_𝑝+𝑟+1⋯𝑢^𝑝−1_𝑝+𝑟+1𝑢^𝑝+𝑞_𝑝+𝑟+1⋯𝑢^{𝑝+𝑞+𝑟−1}_𝑝+𝑟+1𝑢^𝑝_𝑝+𝑟+1⋯𝑢^{𝑝+𝑞−1}_𝑝+𝑟+1⋮⋮⋮⋮⋮⋮⋮1𝑢_{𝑝+𝑞+𝑟}⋯𝑢^𝑝−1_{𝑝+𝑞+𝑟}𝑢^𝑝+𝑞_{𝑝+𝑞+𝑟}⋯𝑢^{𝑝+𝑞+𝑟−1}_{𝑝+𝑞+𝑟}𝑢^𝑝_{𝑝+𝑞+𝑟}⋯𝑢^{𝑝+𝑞−1}_{𝑝+𝑞+𝑟}. (A.4)By appealing to an expansion by blocks of a determinant, it may be seen that 𝑈_𝑝𝑞𝑟(𝑢₁,…,𝑢_𝑝+𝑟) is the cofactor of the “south-east” block of the above determinant. Since the product of, for example, the diagonal terms of this last block is 𝑢^𝑝_𝑝+𝑟+1𝑢^𝑝+1_𝑝+𝑟+2⋯𝑢^{𝑝+𝑞−1}_{𝑝+𝑞+𝑟}, the determinant 𝑈_𝑝𝑞𝑟(𝑢₁,…,𝑢_𝑝+𝑟) is also the coefficient of 𝑢^𝑝_𝑝+𝑟+1𝑢^𝑝+1_𝑝+𝑟+2⋯𝑢^{𝑝+𝑞−1}_{𝑝+𝑞+𝑟} in 𝑉_{𝑝+𝑞+𝑟}(𝑢₁,…,𝑢_{𝑝+𝑞+𝑟}). Now, let us expand 𝑉_{𝑝+𝑞+𝑟}(𝑢₁,…,𝑢_{𝑝+𝑞+𝑟}): 𝑉_{𝑝+𝑞+𝑟}𝑢₁,…,𝑢_{𝑝+𝑞+𝑟}=_{1≤𝑖<𝑗≤𝑝+𝑞+𝑟}𝑢_𝑗−𝑢_𝑖=₁₂₃ (A.5) with ₁=_{1≤𝑖<𝑗≤𝑝+𝑟}𝑢_𝑗−𝑢_𝑖,₂=_{1≤𝑖≤𝑝+𝑟𝑝+𝑟+1≤𝑗≤𝑝+𝑞+𝑟}𝑢_𝑗−𝑢_𝑖,₃=_{𝑝+𝑟+1≤𝑖<𝑗≤𝑝+𝑞+𝑟}𝑢_𝑗−𝑢_𝑖. (A.6) We have that ₂=^{𝑝+𝑞+𝑟}_{𝑗=𝑝+𝑟+1}𝑢_𝑗−𝑢₁⋯𝑢_𝑗−𝑢_𝑝+𝑟=^{𝑝+𝑞+𝑟}_{𝑗=𝑝+𝑟+1}^𝑝+𝑟_𝑘=0(−1)^{𝑝+𝑟−𝑘}𝑠_{𝑝+𝑟−𝑘}𝑢^𝑘_𝑗=_{0≤𝑘1,…,𝑘𝑞≤𝑝+𝑟}(−1)^{𝑞(𝑝+𝑟)−𝑘1−⋯−𝑘𝑞}×𝑠_{𝑝+𝑟−𝑘1}⋯𝑠_{𝑝+𝑟−𝑘𝑞}𝑢^𝑘1_𝑝+𝑟+1⋯𝑢^𝑘𝑞_{𝑝+𝑞+𝑟},₃=𝑉_𝑞𝑢_𝑝+𝑟+1,…,𝑢_{𝑝+𝑞+𝑟}=_{𝜍∈𝔖𝑞}𝜖(𝜍)𝑢^𝜍(1)−1_𝑝+𝑟+1⋯𝑢^{𝜍(𝑞)−1}_{𝑝+𝑞+𝑟}=_{0≤ℓ1,…,ℓ𝑞ℓ≤𝑞−11,…,ℓ𝑞alldifferent}𝜖ℓ₁…,ℓ_𝑞𝑢^ℓ1_𝑝+𝑟+1⋯𝑢^ℓ𝑞_{𝑝+𝑞+𝑟}. (A.7) The symbol 𝔖_𝑞 in the above sum denotes the set of the permutations of the numbers 1,2,…,𝑞, 𝜖(𝜍) is the signature of the permutation 𝜍 and 𝜖(ℓ₁…,ℓ_𝑞)∈{−1,+1} is the signature of the permutation mapping 1,…,𝑞 into ℓ₁…,ℓ_𝑞. The product ∏₂∏₃ is given by (−1)^𝑞(𝑝+1)_{0≤𝑘1,…,𝑘𝑞≤𝑝+𝑟0≤ℓ1,…,ℓ𝑞ℓ≤𝑞−11,…,ℓ𝑞alldifferent}(−1)^{𝑘1+⋯+𝑘𝑞}𝜖ℓ₁…,ℓ_𝑞×𝑠_{𝑝+𝑟−𝑘1}⋯𝑠_{𝑝+𝑟−𝑘𝑞}𝑢^𝑘1+ℓ1_𝑝+𝑟+1⋯𝑢^{𝑘𝑞+ℓ𝑞}_{𝑝+𝑞+𝑟}. (A.8) For obtaining the coefficient of 𝑢^𝑝_𝑝+𝑟+1𝑢^𝑝+1_𝑝+𝑟+2⋯𝑢^{𝑝+𝑞−1}_{𝑝+𝑞+𝑟}, we only keep in the foregoing sum the indices 𝑘₁,…,𝑘_𝑞,ℓ₁,…,ℓ_𝑞 such that 𝑘₁+ℓ₁=𝑝, 𝑘₂+ℓ₂=𝑝+1,…, 𝑘_𝑞+ℓ_𝑞=𝑝+𝑞−1 and 0≤𝑘₁,…,𝑘_𝑞≤𝑝+𝑟, 0≤ℓ₁,…,ℓ_𝑞≤𝑞−1, the indices ℓ₁,…,ℓ_𝑞 being all distinct. This gives that ₂₃=_{for∶()𝑖∈1,…,𝑞𝑖−𝑟−1∨0≤ℓ𝑖≤∧ℓ𝑝+𝑖−1𝑞−11,…,ℓ𝑞alldifferent}𝜖ℓ₁…,ℓ_𝑞×𝑠_ℓ1+𝑟𝑠_{ℓ2+𝑟−1}⋯𝑠_{ℓ𝑞+𝑟−𝑞+1}. (A.9)

Finally, we can observe that the foregoing sum is nothing but the expansion of the determinant 𝒮_𝑝𝑞𝑟=𝑠_𝑟𝑠_𝑟−1⋯𝑠_{𝑟−𝑞+1}𝑠_𝑟+1𝑠_𝑟⋯𝑠_{𝑟−𝑞+2}𝑠⋮⋮⋮_{𝑟+𝑞−1}𝑠_{𝑟+𝑞−2}⋯𝑠_𝑟. (A.10) As a result, we obtain the result below.

Proposition A.1. The determinant 𝑈_𝑝𝑞𝑟(𝑢₁,…,𝑢_𝑝+𝑟) admits the following expression:𝑈_𝑝𝑞𝑟𝑢₁,…,𝑢_𝑝+𝑟=_{1≤𝑖<𝑗≤𝑝+𝑟}𝑢_𝑗−𝑢_𝑖×^{𝑠𝑟𝑢1,…,𝑢𝑝+𝑟𝑠𝑟−1𝑢1,…,𝑢𝑝+𝑟⋯𝑠𝑟−𝑞+1𝑢1,…,𝑢𝑝+𝑟𝑠𝑟+1𝑢1,…,𝑢𝑝+𝑟𝑠𝑟𝑢1,…,𝑢𝑝+𝑟⋯𝑠𝑟−𝑞+2𝑢1,…,𝑢𝑝+𝑟}_{𝑠⋮⋮⋮𝑟+𝑞−1𝑢1,…,𝑢𝑝+𝑟𝑠𝑟+𝑞−2𝑢1,…,𝑢𝑝+𝑟⋯𝑠𝑟𝑢1,…,𝑢𝑝+𝑟}. (A.11)

Let now 𝑈_{𝑝𝑞𝑟,ℓ}^{𝑢1,…,𝑢𝑝+𝑟}_{𝛼1,…,𝛼𝑝+𝑟} be the determinant deduced from 𝑈_𝑝𝑞𝑟(𝑢₁,…,𝑢_𝑝+𝑟) by replacing one column by a general column 𝛼₁,…,𝛼_𝑝+𝑟, that is, the determinant given, if 0≤ℓ≤𝑝−1, by 1𝑢₁⋯𝑢₁^ℓ−1𝛼₁𝑢₁^ℓ+1⋯𝑢₁^𝑝−1𝑢₁^𝑝+𝑞⋯𝑢₁^{𝑝+𝑞+𝑟−1}⋮⋮⋮⋮⋮⋮⋮⋮1𝑢_𝑝+𝑟⋯𝑢^ℓ−1_𝑝+𝑟𝛼_𝑝+𝑟𝑢^ℓ+1_𝑝+𝑟⋯𝑢^𝑝−1_𝑝+𝑟𝑢^𝑝+𝑞_𝑝+𝑟⋯𝑢^{𝑝+𝑞+𝑟−1}_𝑝+𝑟, (A.12)

and, if 𝑝+𝑞≤ℓ≤𝑝+𝑞+𝑟−1, by 1𝑢₁⋯𝑢₁^𝑝−1𝑢₁^𝑝+𝑞⋯𝑢₁^ℓ−1𝛼₁𝑢₁^ℓ+1⋯𝑢₁^{𝑝+𝑞+𝑟−1}⋮⋮⋮⋮⋮⋮⋮⋮1𝑢_𝑝+𝑟⋯𝑢^𝑝−1_𝑝+𝑟𝑢^𝑝+𝑞_𝑝+𝑟⋯𝑢^ℓ−1_𝑝+𝑟𝛼_𝑝+𝑟𝑢^ℓ+1_𝑝+𝑟⋯𝑢^{𝑝+𝑞+𝑟−1}_𝑝+𝑟. (A.13)

We have that 𝑈_{𝑝𝑞𝑟,ℓ}𝑢₁,…,𝑢_𝑝+𝑟𝛼₁,…,𝛼_𝑝+𝑟=^𝑝+𝑟_𝑘=1𝛼_𝑘𝑈_{𝑝𝑞𝑟,𝑘ℓ}𝑢₁,…,𝑢_𝑘−1,𝑢_𝑘+1,…,𝑢_𝑝+𝑟, (A.14)where 𝑈_{𝑝𝑞𝑟,𝑘ℓ}(𝑢₁,…,𝑢_𝑘−1,𝑢_𝑘+1,…,𝑢_𝑝+𝑟) is the determinant given, if 0≤ℓ≤𝑝−1, by 1𝑢₁⋯𝑢₁^ℓ−10𝑢₁^ℓ+1⋯𝑢₁^𝑝−1𝑢₁^𝑝+𝑞⋯𝑢₁^{𝑝+𝑞+𝑟−1}⋮⋮⋮⋮⋮⋮⋮⋮1𝑢_𝑘−1⋯𝑢^ℓ−1_𝑘−10𝑢^ℓ+1_𝑘−1⋯𝑢^𝑝−1_𝑘−1𝑢^𝑝+𝑞_𝑘−1⋯𝑢^{𝑝+𝑞+𝑟−1}_𝑘−11𝑢_𝑘⋯𝑢_𝑘^ℓ−11𝑢_𝑘^ℓ+1⋯𝑢_𝑘^𝑝−1𝑢_𝑘^𝑝+𝑞⋯𝑢_𝑘^{𝑝+𝑞+𝑟−1}1𝑢_𝑘+1⋯𝑢^ℓ−1_𝑘+10𝑢^ℓ+1_𝑘+1⋯𝑢^𝑝−1_𝑘+1𝑢^𝑝+𝑞_𝑘+1⋯𝑢^{𝑝+𝑞+𝑟−1}_𝑘+1⋮⋮⋮⋮⋮⋮⋮⋮1𝑢_𝑝+𝑟⋯𝑢^ℓ−1_𝑝+𝑟0𝑢^ℓ+1_𝑝+𝑟⋯𝑢^𝑝−1_𝑝+𝑟𝑢^𝑝+𝑞_𝑝+𝑟⋯𝑢^{𝑝+𝑞+𝑟−1}_𝑝+𝑟, (A.15)

and, if 𝑝+𝑞≤ℓ≤𝑝+𝑞+𝑟−1, by1𝑢₁⋯𝑢₁^𝑝−1𝑢₁^𝑝+𝑞⋯𝑢₁^ℓ−10𝑢₁^ℓ+1⋯𝑢₁^{𝑝+𝑞+𝑟−1}⋮⋮⋮⋮⋮⋮⋮⋮1𝑢_𝑘−1⋯𝑢^𝑝−1_𝑘−1𝑢^𝑝+𝑞_𝑘−1⋯𝑢^ℓ−1_𝑘−10𝑢^ℓ+1_𝑘−1⋯𝑢^{𝑝+𝑞+𝑟−1}_𝑘−11𝑢_𝑘⋯𝑢_𝑘^𝑝−1𝑢_𝑘^𝑝+𝑞⋯𝑢_𝑘^ℓ−11𝑢_𝑘^ℓ+1⋯𝑢_𝑘^{𝑝+𝑞+𝑟−1}1𝑢_𝑘+1⋯𝑢^𝑝−1_𝑘+1𝑢^𝑝+𝑞_𝑘+1⋯𝑢^ℓ−1_𝑘+10𝑢^ℓ+1_𝑘+1⋯𝑢^{𝑝+𝑞+𝑟−1}_𝑘+1⋮⋮⋮⋮⋮⋮⋮⋮1𝑢_𝑝+𝑟⋯𝑢^𝑝−1_𝑝+𝑟𝑢^𝑝+𝑞_𝑝+𝑟⋯𝑢^ℓ−1_𝑝+𝑟0𝑢^ℓ+1_𝑝+𝑟⋯𝑢^{𝑝+𝑞+𝑟−1}_𝑝+𝑟. (A.16)

In fact, 𝑈_{𝑝𝑞𝑟,𝑘ℓ}(𝑢₁,…,𝑢_𝑘−1,𝑢_𝑘+1,…,𝑢_𝑝+𝑟) is the coefficient of 𝑢^ℓ_𝑘 in 𝑈_𝑝𝑞𝑟(𝑢₁,…,𝑢_𝑝+𝑟). Let us introduce 𝑠_𝑘,0∶=𝑠_𝑘,0(𝑢₁,…,𝑢_𝑘−1,𝑢_𝑘+1,…,𝑢_𝑝+𝑟)=1, 𝑠_𝑘,𝑚∶=𝑠_𝑘,𝑚(𝑢₁,…,𝑢_𝑘−1,𝑢_𝑘+1,…,𝑢_𝑝+𝑟)=0 for any integer 𝑚 such that 𝑚≤−1 or 𝑚≥𝑝+𝑟 and, for 𝑚∈{1,…,𝑝+𝑟−1}, 𝑠_𝑘,𝑚∶=𝑠_𝑘,𝑚𝑢₁,…,𝑢_𝑘−1,𝑢_𝑘+1,…,𝑢_𝑝+𝑟=_{1≤𝑖1<⋯<𝑖𝑚𝑖≤𝑝+𝑟1,…,𝑖𝑚≠𝑘}𝑢_𝑖1⋯𝑢_𝑖𝑚. (A.17)

We need to isolate 𝑢_𝑘 in ∏_{1≤𝑖<𝑗≤𝑝+𝑟}(𝑢_𝑗−𝑢_𝑖) and 𝒮_𝑝𝑞𝑟. First, we write that _{1≤𝑖<𝑗≤𝑝+𝑟}𝑢_𝑗−𝑢_𝑖=(−1)^{𝑝+𝑟−𝑘}_{1≤𝑖<𝑗≤𝑝+𝑟𝑖,𝑗≠𝑘}𝑢_𝑗−𝑢_𝑖×_{1≤𝑖≤𝑝+𝑟𝑗≠𝑘}𝑢_𝑘−𝑢_𝑖=(−1)^𝑘−1_{1≤𝑖<𝑗≤𝑝+𝑟𝑖,𝑗≠𝑘}𝑢_𝑗−𝑢_𝑖^{𝑝+𝑟−1}_𝑚=0(−1)^𝑚𝑠_{𝑘,𝑝+𝑟−𝑚−1}𝑢^𝑚_𝑘. (A.18)

Second, by isolating 𝑢_𝑘 in 𝑠_𝑚 according to 𝑠_𝑚=𝑠_𝑘,𝑚+𝑢_𝑘𝑠_{𝑘,𝑚−1}, we get that the determinant 𝒮_pqr can be rewritten as 𝑠_𝑘,𝑟+𝑢_𝑘𝑠_{𝑘,𝑟−1}𝑠_{𝑘,𝑟−1}+𝑢_𝑘𝑠_{𝑘,𝑟−2}⋯𝑠_{𝑘,𝑟−𝑞+1}+𝑢_𝑘𝑠_{𝑘,𝑟−𝑞}𝑠_𝑘,𝑟+1+𝑢_𝑘𝑠_𝑘,𝑟𝑠_𝑘,𝑟+𝑢_𝑘𝑠_{𝑘,𝑟−1}⋯𝑠_{𝑘,𝑟−𝑞+2}+𝑢_𝑘𝑠_{𝑘,𝑟−𝑞+1}𝑠⋮⋮⋮_{𝑘,𝑟+𝑞−1}+𝑢_𝑘𝑠_{𝑘,𝑟+𝑞−2}𝑠_{𝑘,𝑟+𝑞−2}+𝑢_𝑘𝑠_{𝑘,𝑟+𝑞−3}⋯𝑠_𝑘,𝑟+𝑢_𝑘𝑠_{𝑘,𝑟−1}. (A.19)

By introducing vectors 𝑉^→_𝑚 with coordinates (𝑠_𝑘,𝑚,𝑠_𝑘,𝑚+1,…,𝑠_{𝑘,𝑚+𝑞−1}) (written as a column), this determinant can be rewritten as 𝒮_𝑝𝑞𝑟=det𝑉^→_𝑟+𝑢𝑉_𝑘^→_𝑟−1,𝑉^→_𝑟−1+𝑢𝑉_𝑘^→_𝑟−2,…,𝑉^→_{𝑟−𝑞+1}+𝑢𝑉_𝑘^→_𝑟−𝑞. (A.20)

By appealing to multilinearity, it is easy to see that 𝒮_𝑝𝑞𝑟=^𝑞_𝑛=0det𝑉^→_𝑟,𝑉^→_𝑟−1,…,𝑉^→_{𝑟−𝑞+𝑛+1},𝑉^→_{𝑟−𝑞+𝑛−1},…,𝑉^→_𝑟−𝑞𝑢^𝑛_𝑘. (A.21) Now, let us multiply the sum lying in (A.18) by (A.21): ^{𝑝+𝑟−1}_𝑚=0(−1)^𝑚𝑠_{𝑘,𝑝+𝑟−𝑚−1}𝑢^𝑚_𝑘×^𝑞_𝑛=0det𝑉^→_𝑟,𝑉^→_𝑟−1,…,𝑉^→_{𝑟−𝑞+𝑛+1},𝑉^→_{𝑟−𝑞+𝑛−1},…,𝑉^→_𝑟−𝑞𝑢^𝑛_𝑘=^{𝑝+𝑞+𝑟−1}_ℓ=0^{(𝑝+𝑟−1)∧ℓ}_{𝑚=0∨ℓ−𝑞}(−1)^𝑚𝑠_{𝑘,𝑝+𝑟−𝑚−1}×det𝑉^→_𝑟,𝑉^→_𝑟−1,…,𝑉^→_{ℓ+𝑟−𝑞−𝑚+1},_→𝑉_{ℓ+𝑟−𝑞−𝑚−1},…,𝑉^→_𝑟−𝑞𝑢^ℓ_𝑘. (A.22)

Recalling the convention that 𝑠_𝑘,𝑚=0 if 𝑚≤−1 or 𝑚≥𝑝+𝑟, the coefficient of 𝑢^ℓ_𝑘 in (A.22) can be written as ^ℓ_{𝑚=ℓ−𝑞}(−1)^𝑚𝑠_{𝑘,𝑝+𝑟−𝑚−1}×det𝑉^→_𝑟,𝑉^→_𝑟−1,…,𝑉^→_{ℓ+𝑟−𝑞−𝑚+1},𝑉^→_{ℓ+𝑟−𝑞−𝑚−1},…,𝑉^→_𝑟−𝑞=(−1)^ℓ𝑞_𝑚=0(−1)^𝑚+𝑞𝑠_{𝑘,𝑝+𝑞+𝑟−ℓ−𝑚−1}×det𝑉^→_𝑟,𝑉^→_𝑟−1,…,𝑉^→_{𝑟−𝑚+1},𝑉^→_{𝑟−𝑚−1},…,𝑉^→_𝑟−𝑞. (A.23)

In this form, we see that the coefficient of 𝑢^ℓ_𝑘 in (A.22) is nothing but the product of (−1)^ℓ by the expansion of the determinant 𝒮_{𝑝𝑞𝑟,𝑘ℓ}=𝑠_𝑘,𝑟𝑠_{𝑘,𝑟−1}⋯𝑠_{𝑘,𝑟−𝑞}𝑠_𝑘,𝑟+1𝑠_𝑘,𝑟⋯𝑠_{𝑘,𝑟−𝑞+1}𝑠⋮⋮⋮_{𝑘,𝑟+𝑞−1}𝑠_{𝑘,𝑟+𝑞−2}⋯𝑠_{𝑘,𝑟−1}𝑠_{𝑘,𝑝+𝑞+𝑟−ℓ−1}𝑠_{𝑘,𝑝+𝑞+𝑟−ℓ−2}⋯𝑠_{𝑘,𝑝+𝑟−ℓ−1}. (A.24)

Proposition A.2. The determinant 𝑈_{𝑝𝑞𝑟,𝑘ℓ}(𝑢₁,…,𝑢_𝑘−1,𝑢_𝑘+1,…,𝑢_𝑝+𝑟) admits the following expression: 𝑈_{𝑝𝑞𝑟,𝑘ℓ}𝑢₁,…,𝑢_𝑘−1,𝑢_𝑘+1,…,𝑢_𝑝+𝑟=(−1)^𝑘+ℓ−1_{1≤𝑖<𝑗≤𝑝+𝑟𝑖,𝑗≠𝑘}𝑢_𝑗−𝑢_𝑖×𝑠_𝑘,𝑟𝑢₁,…,𝑢_𝑘−1,𝑢_𝑘+1,…,𝑢_𝑝+𝑟⋯𝑠_{𝑘,𝑟−𝑞}𝑢₁,…,𝑢_𝑘−1,𝑢_𝑘+1,…,𝑢_𝑝+𝑟𝑠_𝑘,𝑟+1𝑢₁,…,𝑢_𝑘−1,𝑢_𝑘+1,…,𝑢_𝑝+𝑟⋯𝑠_{𝑘,𝑟−𝑞+1}𝑢₁,…,𝑢_𝑘−1,𝑢_𝑘+1,…,𝑢_𝑝+𝑟𝑠⋮⋮_{𝑘,𝑟+𝑞−1}𝑢₁,…,𝑢_𝑘−1,𝑢_𝑘+1,…,𝑢_𝑝+𝑟⋯𝑠_{𝑘,𝑟−1}𝑢₁,…,𝑢_𝑘−1,𝑢_𝑘+1,…,𝑢_𝑝+𝑟𝑠_{𝑘,𝑝+𝑞+𝑟−ℓ−1}𝑢₁,…,𝑢_𝑘−1,𝑢_𝑘+1,…,𝑢_𝑝+𝑟⋯𝑠_{𝑘,𝑝+𝑟−ℓ−1}𝑢₁,…,𝑢_𝑘−1,𝑢_𝑘+1,…,𝑢_𝑝+𝑟. (A.25)

As a consequence of Propositions A.1 and A.2, we get the result below. Set 𝑝_𝑘∶=𝑝_𝑘𝑢₁,…,𝑢_𝑝+𝑟=_{1≤𝑖≤𝑝+𝑟𝑖≠𝑘}𝑢_𝑘−𝑢_𝑖. (A.26)

Proposition A.3. Let 𝑝,𝑞,𝑟 be positive integers, let 𝑢₁,…,𝑢_𝑝+𝑟 be distinct complex numbers, and let 𝛼₁,…,𝛼_𝑝+𝑟 be complex numbers. Set 𝐼={0,…,𝑝−1}∪{𝑝+𝑞,…,𝑝+𝑞+𝑟−1}. The solution of the system ∑_ℓ∈𝐼𝑢^ℓ_𝑘𝑥_ℓ=𝛼_𝑘, 1≤𝑘≤𝑝+𝑟, or, more explicitly, 𝑥₀+𝑢₁𝑥₁+⋯+𝑢₁^𝑝−1𝑥_𝑝−1+𝑢₁^𝑝+𝑞𝑥_𝑝+𝑞+⋯+𝑢₁^{𝑝+𝑞+𝑟−1}𝑥_{𝑝+𝑞+𝑟−1}=𝛼₁⋮𝑥₀+𝑢_𝑝+𝑟𝑥₁+⋯+𝑢^𝑝−1_𝑝+𝑟𝑥_𝑝−1+𝑢^𝑝+𝑞_𝑝+𝑟𝑥_𝑝+𝑞+⋯+𝑢^{𝑝+𝑞+𝑟−1}_𝑝+𝑟𝑥_{𝑝+𝑞+𝑟−1}=𝛼_𝑝+𝑟 (A.27) is given by 𝑥_ℓ=(−1)^{ℓ+𝑝+𝑟−1}𝒮_𝑝𝑞𝑟𝑢₁,…,𝑢_𝑝+𝑟×^𝑝+𝑟_𝑘=1𝛼_𝑘𝒮_{𝑝𝑞𝑟,𝑘ℓ}𝑢₁,…,𝑢_𝑘−1,𝑢_𝑘+1,…,𝑢_𝑝+𝑟𝑝_𝑘𝑢₁,…,𝑢_𝑝+𝑟,ℓ∈𝐼. (A.28)

Proof. Cramer’s formulae yield that 𝑥_ℓ=𝑈_{𝑝𝑞𝑟,ℓ}^{𝑢1,…,𝑢𝑝+𝑟}_{𝛼1,…,𝛼𝑝+𝑟}𝑈_𝑝𝑞𝑟𝑢₁,…,𝑢_𝑝+𝑟=^𝑝+𝑟_𝑘=1𝛼_𝑘𝑈_{𝑝𝑞𝑟,𝑘ℓ}𝑢₁,…,𝑢_𝑘−1,𝑢_𝑘+1,…,𝑢_𝑝+𝑟𝑈_𝑝𝑞𝑟𝑢₁,…,𝑢_𝑝+𝑟. (A.29) By using the factorisations provided by Propositions A.1 and A.2, namely, 𝑈_𝑝𝑞𝑟𝑢₁,…,𝑢_𝑝+𝑟=𝑉_𝑝+𝑟𝑢₁,…,𝑢_𝑝+𝑟𝒮_𝑝𝑞𝑟𝑢₁,…,𝑢_𝑝+𝑟=(−1)^{𝑝+𝑟−𝑘}𝑝_𝑘𝑢₁,…,𝑢_𝑝+𝑟×𝑉_{𝑝+𝑟−1}𝑢₁,…,𝑢_𝑘−1,𝑢_𝑘+1,…,𝑢_𝑝+𝑟𝒮_𝑝𝑞𝑟𝑢₁,…,𝑢_𝑝+𝑟,𝑈_{𝑝𝑞𝑟,𝑘ℓ}𝑢₁,…,𝑢_𝑘−1,𝑢_𝑘+1,…,𝑢_𝑝+𝑟=(−1)^𝑘+ℓ−1𝑉_{𝑝+𝑟−1}𝑢₁,…,𝑢_𝑘−1,𝑢_𝑘+1,…,𝑢_𝑝+𝑟×𝒮_{𝑝𝑞𝑟,𝑘ℓ}𝑢₁,…,𝑢_𝑘−1,𝑢_𝑘+1,…,𝑢_𝑝+𝑟, (A.30) we immediately get (A.28).

A.2. A Combinatoric Identity

Lemma A.4. The following identity holds for any positive integers 𝛼, 𝛽, 𝑛: ^𝑛_𝑘=0(−1)^𝑘𝑛𝑘(𝑘+𝛼)!!=𝑘+𝛽𝛼!𝛽+𝑛!𝛽−𝛼+𝑛−1_𝑛. (A.31) It can be rewritten as ^𝑛_𝑘=0(−1)^𝑘𝑛𝑘=()𝑘+𝛼𝑘+𝛽−1^𝑛𝛼𝛽+𝑛𝑖𝑓𝛼≥𝛽,_𝑛𝑘=0(−1)^𝑘(^𝑛_𝑘)_{𝑘+𝛽𝑘+𝛼}=𝛽−𝛼/𝛽−𝛼+𝑛_{𝛼𝛽+𝑛}𝑖𝑓𝛼<𝛽. (A.32)

Proof. Suppose first that 𝛼≥𝛽. Noticing that (𝑘+𝛼)!!=𝑘+𝛽d^𝛼−𝛽d𝑥^𝛼−𝛽𝑥^𝑘+𝛼_𝑥=1, (A.33) we immediately get that ^𝑛_𝑘=0(−1)^𝑘𝑛𝑘(𝑘+𝛼)!!=𝑘+𝛽d^𝛼−𝛽d𝑥^{𝛼−𝛽𝑛}_𝑘=0(−1)^𝑘𝑛𝑘𝑥^𝑘+𝛼_𝑥=1=d^𝛼−𝛽d𝑥^𝛼−𝛽𝑥^𝛼(1−𝑥)^𝑛_𝑥=1. (A.34) We expand the last displayed derivative by using Leibniz rule: d^𝛼−𝛽d𝑥^𝛼−𝛽𝑥^𝛼(1−𝑥)^𝑛_𝑥=1=^𝛼−𝛽_𝑘=0𝑘𝛼−𝛽d^{𝛼−𝛽−𝑘}d𝑥^{𝛼−𝛽−𝑘}𝑥^𝛼_𝑥=1d^𝑘d𝑥^𝑘(1−𝑥)^𝑛_𝑥=1. (A.35) Since (d^𝑘/d𝑥^𝑘)((1−𝑥)^𝑛)|_𝑥=1=(−1)^𝑛𝛿_𝑛𝑘𝑛!, we have that (d^𝛼−𝛽/d𝑥^𝛼−𝛽)(𝑥^𝛼(1−𝑥)^𝑛)|_𝑥=1=0 if 𝛼−𝛽<𝑛, and, if 𝛼−𝛽≥𝑛, d^𝛼−𝛽d𝑥^𝛼−𝛽𝑥^𝛼(1−𝑥)^𝑛_𝑥=1=(−1)^𝑛𝑛𝑛!𝛼−𝛽d^{𝛼−𝛽−𝑛}d𝑥^{𝛼−𝛽−𝑛}𝑥^𝛼_𝑥=1=(−1)^𝑛𝛼!𝛼−𝛽!𝛼−𝛽−𝑛!𝛽+𝑛! (A.36) which coincides with the announced result. Second, suppose that 𝛼<𝛽. Noticing that (𝑘+𝛼)!!=1𝑘+𝛽𝛽−𝛼−1!¹₀𝑥^𝑘+𝛼(1−𝑥)^{𝛽−𝛼−1}d𝑥, (A.37) we get that ^𝑛_𝑘=0(−1)^𝑘𝑛𝑘(𝑘+𝛼)!!=1𝑘+𝛽!×𝛽−𝛼−1_10𝑛𝑘=0(−1)^𝑘𝑛𝑘𝑥^𝑘𝑥^𝛼(1−𝑥)^{𝛽−𝛼−1}d𝑥=1𝛽−𝛼−1!¹₀𝑥^𝛼(1−𝑥)^{𝛽−𝛼+𝑛−1}d𝑥=𝛼!𝛽−𝛼+𝑛−1!𝛽−𝛼−1!𝛽+𝑛! (A.38) which coincides with the announced result.

A.3. Some Matrices

Let 𝛼,𝛽∈ℕ such that 𝛼>𝛽≥𝑁 and set 𝐀=𝑗+𝛼𝑖+𝑁^{0≤𝑖,𝑗≤𝑁−1}𝛽,𝐁=𝑖+𝑁^{0≤𝑖≤𝑁−1} (A.39) with the convention of settings ^𝑖_𝑗=0 if 𝑖>𝑗. These matrices have been used for solving systems (273) and (274) with the choices 𝛼=𝑏−𝑎+𝑁−1 and 𝛽=𝑁−𝑎−1. The aim of this section is to compute the product of the inverse of 𝐀 by 𝐁, namely, 𝐀⁻¹𝐁. For this, we use Gauss elimination. The result is displayed in Theorem A.8. The calculations are quite technical, so we perform them progressively, the intermediate steps being stated in several lemmas (Lemmas A.4, A.5, A.6, and A.7).

Lemma A.5. The matrix 𝐀 can be decomposed into 𝐀=𝐋𝐔⁻¹, where the matrices 𝐔 and 𝐋 are given by 𝐔=(−1)^𝑖+𝑗𝟙_{{𝑖≤𝑗}}𝑗𝑖𝑗+𝛼_𝑁(𝑖+𝛼)_𝑁^{0≤𝑖,𝑗≤𝑁−1},𝐋=𝟙_{{𝑖≥𝑗}}𝑗+𝛼(𝑖)𝑖+𝑁^𝑗𝑗+𝛼−𝑁_𝑗^{0≤𝑖,𝑗≤𝑁−1}. (A.40) The regular matrices 𝐔 and 𝐋, are respectively, upper and lower triangular.

Proof. We begin by detailing the algorithm providing the matrix 𝐔. Call 𝐂₀⁽⁰⁾,𝐂₁⁽⁰⁾,…,𝐂⁽⁰⁾_𝑁−1 the columns of 𝐀, that is, for 𝑗∈{0,…,𝑁−1}, 𝐂_𝑗⁽⁰⁾=𝑗+𝛼𝑖+𝑁^{0≤𝑖≤𝑁−1}. (A.41) Apply to them, except for 𝐂₀⁽⁰⁾, the transformation defined, for 𝑗∈{1,…,𝑁−1}, by 𝐂_𝑗⁽¹⁾=𝐂_𝑗⁽⁰⁾−𝑗+𝛼𝐂𝑗+𝛼−𝑁^(0)𝑗−1. (A.42) The 𝐂₀⁽⁰⁾,𝐂₁⁽¹⁾,…,𝐂⁽¹⁾_𝑁−1 are the columns of a new matrix 𝐋₁. Actually, this transformation corresponds to a matrix multiplication acting on 𝐀: 𝐋₁=𝐀𝐔₁ with 𝐔₁=1−𝛼+10𝛼+1−𝑁001−𝛼+2⋮0𝛼+2−𝑁⋮01⋱0⋮⋱−𝛼+𝑁−10𝛼−1.001 (A.43)
Simple computations show that 𝐋₁=𝛿_𝑗0+𝟙_{𝑗≥1}𝑖𝑗+𝛼−𝑁𝑗+𝛼𝑖+𝑁^{0≤𝑖,𝑗≤𝑁−1}=𝑗+𝛼(𝑖)𝑖+𝑁^𝑗∧1𝑗+𝛼−𝑁_𝑗∧1^{0≤𝑖,𝑗≤𝑁−1}. (A.44)
Next, apply the second transformation to the columns of 𝐋₁ except for 𝐂₀⁽⁰⁾ and 𝐂₁⁽¹⁾, defined, for 𝑗∈{2,…,𝑁−1}, by 𝐂_𝑗⁽²⁾=𝐂_𝑗⁽¹⁾−𝑗+𝛼𝐂𝑗+𝛼−𝑁^(1)𝑗−1=𝐂_𝑗⁽⁰⁾−2𝑗+𝛼𝐂𝑗+𝛼−𝑁^(0)𝑗−1+𝑗+𝛼𝑗+𝛼−1𝐂𝑗+𝛼−𝑁𝑗+𝛼−𝑁−1^(1)𝑗−2. (A.45)
The 𝐂₀⁽⁰⁾,𝐂₁⁽¹⁾,𝐂₂⁽²⁾,…,𝐂⁽²⁾_𝑁−1 are the columns of a new matrix 𝐋₂=𝐀𝐔₂, where𝐔₂=1−𝛼+1𝛼+1−𝑁(𝛼+2)(𝛼+1)()0𝛼+2−𝑁)(𝛼+1−𝑁01−2𝛼+2𝛼+2−𝑁⋱⋮00()1⋱𝛼+𝑁−1)(𝛼+𝑁−2(𝛼−1)(𝛼−2)⋮⋮⋱−2𝛼+𝑁−1𝛼−100.01 (A.46)Straightforward algebra yields that 𝐋₂=𝛿_𝑗0+𝑖𝛿𝑗+𝛼−𝑁^𝑗1+𝟙_{𝑗≥2}𝑖(𝑖−1)×𝑗+𝛼−𝑁𝑗+𝛼−𝑁−1𝑗+𝛼𝑖+𝑁_{0≤𝑖,𝑗≤𝑁−1}=𝑗+𝛼(𝑖)𝑖+𝑁^𝑗∧2𝑗+𝛼−𝑁_𝑗∧2^{0≤𝑖,𝑗≤𝑁−1}. (A.47) This method can be recursively extended: apply the 𝑘th transformation (1≤𝑘≤𝑁−1) defined, for 𝑗∈{𝑘,…,𝑁−1}, by 𝐂_𝑗^(𝑘)=𝐂_𝑗^(𝑘−1)−𝑗+𝛼𝐂𝑗+𝛼−𝑁^{(𝑘−1)𝑗−1}=𝐂_𝑗^(𝑘−2)−2𝑗+𝛼𝐂𝑗+𝛼−𝑁^{(𝑘−2)𝑗−1}+𝑗+𝛼𝑗+𝛼−1𝐂𝑗+𝛼−𝑁𝑗+𝛼−𝑁−1^{(𝑘−2)𝑗−2}⋮=𝐂_𝑗⁽⁰⁾−𝑘1𝑗+𝛼𝐂𝑗+𝛼−𝑁^(0)𝑗−1+𝑘2𝑗+𝛼𝑗+𝛼−1𝐂𝑗+𝛼−𝑁𝑗+𝛼−𝑁−1^(0)𝑗−2+⋯+(−1)^𝑘𝑘𝑘×𝑗+𝛼𝑗+𝛼−1⋯𝑗+𝛼−𝑘+1𝑗+𝛼−𝑁𝑗+𝛼−𝑁−1⋯𝑗+𝛼−𝑘−𝑁+1×𝐂⁽⁰⁾_𝑗−𝑘=^𝑘_ℓ=0(−1)^ℓ𝑘ℓ𝑗+𝛼_ℓ𝑗+𝛼−𝑁_ℓ𝐂⁽⁰⁾_𝑗−ℓ=^𝑘_ℓ=0(−1)^ℓ𝑘ℓ𝑗+𝛼_𝑁𝑗+𝛼−ℓ_𝑁𝐂⁽⁰⁾_𝑗−ℓ=^𝑗_{ℓ=𝑗−𝑘}(−1)^𝑗−ℓ𝑘𝑗−ℓ𝑗+𝛼_𝑗−ℓ𝑗+𝛼−𝑁_𝑗−ℓ𝐂_ℓ⁽⁰⁾=^𝑗_{ℓ=𝑗−𝑘}(−1)^𝑗−ℓ𝑘𝑗−ℓ𝑗+𝛼_𝑁(ℓ+𝛼)_𝑁𝐂_ℓ⁽⁰⁾. (A.48) In particular, 𝐂_𝑘^(𝑘)=^𝑘_ℓ=0(−1)^𝑘−ℓ𝑘ℓ(𝑘+𝛼)_𝑁(ℓ+𝛼)_𝑁𝐂_ℓ⁽⁰⁾. (A.49)
The 𝐂₀⁽⁰⁾,𝐂₁⁽¹⁾,𝐂₂⁽²⁾,…,𝐂_𝑘^(𝑘),𝐂^(𝑘)_𝑘+1,…,𝐂^(𝑘)_𝑁−1 are the columns of the 𝑘th matrix 𝐋_𝑘=𝐀𝐔_𝑘 with 𝐔_𝑘=[𝐔^′_𝑘⋮𝐔_𝑘^′′], where 𝐔^′_𝑘 is the matrix1−𝛼+1()𝛼+1−𝑁𝛼+2)(𝛼+1()()𝛼+2−𝑁)(𝛼+1−𝑁−1^𝑘−1()⋯()𝑘−1𝑘−1𝛼+𝑘−1𝛼+1(𝛼+𝑘−𝑁−1)⋯(𝛼+1−𝑁)01−2𝛼+2⋱()𝛼+2−𝑁−1^𝑘−2()⋯()𝑘−1𝑘−2𝛼+𝑘−1𝛼+2()⋯()()𝛼+𝑘−𝑁−1𝛼+2−𝑁001⋱⋱−1^𝑘−3()⋯()𝑘−1𝑘−3𝛼+𝑘−1𝛼+3()⋯()2()𝛼+𝑘−𝑁−1𝛼+3−𝑁⋮⋮⋱⋱⋱⋮⋮⋮⋱⋱𝑘−1𝛼+𝑘−1)(𝛼+𝑘−2()1𝛼+𝑘−𝑁−1)(𝛼+𝑘−𝑁−2⋮⋮⋱−𝑘−1𝛼+𝑘−10𝛼+𝑘−𝑁−100𝑘−100⋯⋯⋯⋯0⋮⋮⋮00⋯⋯⋯⋯0 (A.50)
and 𝐔_𝑘^′′ is the matrix^{()−1𝑘𝑘𝑘()⋯()𝛼+𝑘𝛼+1()⋯()⋮()𝛼+𝑘−𝑁𝛼+1−𝑁00−1𝑘𝑘𝑘()⋯()𝛼+𝑘+1𝛼+2()⋯()⋮𝑘2()𝛼+𝑘−𝑁+1𝛼+2−𝑁𝛼+𝑘)(𝛼+𝑘−1()−𝑘1𝛼+𝑘−𝑁)(𝛼+𝑘−𝑁−1⋮⋱0𝛼+𝑘𝑘2()𝛼+𝑘−𝑁𝛼+𝑘+1)(𝛼+𝑘()()𝛼+𝑘−𝑁+1)(𝛼+𝑘−𝑁−1𝑘𝑘𝑘()⋯()𝛼+𝑁−1𝛼+𝑁−𝑘()⋯()}_{𝑘0−𝑘1}^{𝛼−1𝛼−𝑘}_{𝛼+𝑘+10𝑘0⋱𝑘2()𝛼+𝑘−𝑁+1⋱⋮𝛼+𝑁−1)(𝛼+𝑁−2()𝑘1𝛼−1)(𝛼−2⋮⋱−𝛼+𝑁−10𝑘0𝛼−1}. (A.51)
The matrices 𝐔^′_𝑘 and 𝐔_𝑘^′′ can be simply written as 𝐔^′_𝑘=(−1)^𝑗−𝑖𝟙_{{𝑖≤𝑗}}𝑗𝑖𝑗+𝛼_𝑁(𝑖+𝛼)_𝑁^{0≤𝑖≤𝑁−1}_{0≤𝑗≤𝑘−1},𝐔_𝑘^′′=(−1)^𝑗−𝑖𝟙_{{𝑖∨𝑘≤𝑗≤𝑖+𝑘}}𝑘𝑗−𝑖𝑗+𝛼_𝑁(𝑖+𝛼)_𝑁^{0≤𝑖≤𝑁−1}_{𝑘≤𝑗≤𝑁−1}. (A.52)
Clever algebra yields that 𝐋_𝑘=𝑗+𝛼(𝑖)𝑖+𝑁^𝑗∧𝑘𝑗+𝛼−𝑁_𝑗∧𝑘^{0≤𝑖,𝑗≤𝑁−1}. (A.53)
We will not prove (A.53); we will only check it below in the case 𝑘=𝑁−1.
We progressively arrive at the last transformation which corresponds to 𝑘=𝑁−1: 𝐂^(𝑁−1)_𝑁−1=𝐂⁽⁰⁾_𝑁−1−1𝑁−1𝑁+𝛼−1𝐂𝛼−1^(0)𝑁−2+2𝑁−1(𝑁+𝛼−1)(𝑁+𝛼−2)()𝐂𝛼−1)(𝛼−2^(0)𝑁−3+(−1)^𝑁−1×()⋯()𝑁−1𝑁−1𝑁+𝛼−1)(𝑁+𝛼−2𝛼+1()⋯()𝐂𝛼−1)(𝛼−2𝛼−𝑁+1⁰⁽⁰⁾=^𝑁−1_ℓ=0(−1)^ℓℓ𝑁−1(𝑁+𝛼−1)_ℓ(𝛼−1)_ℓ𝐂⁽⁰⁾_{𝑁−1−ℓ}=^𝑁−1_ℓ=0(−1)^{𝑁−1−ℓ}ℓ×()𝑁−1𝑁+𝛼−1_{𝑁−ℓ−1}(𝛼−1)_{𝑁−ℓ−1}𝐂_ℓ⁽⁰⁾. (A.54)
The 𝐂₀⁽⁰⁾,𝐂₁⁽¹⁾,𝐂₂⁽²⁾,…,𝐂^(𝑁−1)_𝑁−1 are the columns of the last matrix given by 𝐋_𝑁−1=𝐀𝐔_𝑁−1, where 𝐔_𝑁−1=(−1)^𝑗−𝑖𝑗𝑖𝑗+𝛼_𝑁(𝑖+𝛼)_𝑁^{0≤𝑖,𝑗≤𝑁−1}. (A.55)
Formula (A.53) gives the following expression for 𝐋_𝑁−1 that will be checked below: 𝐋_𝑁−1=𝑗+𝛼(𝑖)𝑖+𝑁^𝑗𝑗+𝛼−𝑁_𝑗^{0≤𝑖,𝑗≤𝑁−1}. (A.56)
Hence, by putting 𝐋=𝐋_𝑁−1 and 𝐔=𝐔_𝑁−1, we see that 𝐋 is a lower triangular matrix and 𝐔 is an upper triangular matrix and we have obtained that 𝐋=𝐀𝐔.
Finally, we directly check the decomposition 𝐋=𝐀𝐔. The generic term of 𝐀𝐔 is ^𝑗_𝑘=0(−1)^𝑗+𝑘𝑗𝑘𝑘+𝛼𝑖+𝑁𝑗+𝛼_𝑁(𝑘+𝛼)_𝑁. (A.57) Observing that 𝑘+𝛼𝑖+𝑁𝑗+𝛼_𝑁(𝑘+𝛼)_𝑁=^{𝑖𝑘+𝛼−𝑁𝑁𝑗+𝛼}_{𝑁𝑖+𝑁}, (A.58) this term can be rewritten as (−1)^𝑗_𝑁^𝑗+𝛼_{𝑁𝑖+𝑁}×^𝑗_𝑘=0(−1)^𝑘𝑗𝑘𝑖.𝑘+𝛼−𝑁 (A.59)
The sum ∑^𝑗_𝑘=0(−1)^𝑘𝑗_𝑘𝑖^{𝑘+𝛼−𝑁} can be explicitly evaluated thanks to Lemma A.4. Its value is (−1)^{𝑗𝛼−𝑁}_𝑖−𝑗. Therefore, we can easily get the generic term of 𝐋, and the proof of Lemma A.5 is finished.

Lemma A.6. The inverse of the matrix 𝐋 is given by 𝐋⁻¹=(−1)^𝑖+𝑗𝟙_{{𝑖≥𝑗}}𝑗+𝑁!(𝑖+𝛼−𝑁)_𝑖+1𝑗!𝑖−𝑗!(𝑖+𝛼)_𝑗+𝑁+1^{0≤𝑖,𝑗≤𝑁−1}. (A.60)

Proof. We simplify the entries of the product 𝟙_{{𝑖≥𝑗}}𝑗+𝛼(𝑖)𝑖+𝑁^𝑗𝑗+𝛼−𝑁_𝑗^{0≤𝑖,𝑗≤𝑁−1}×(−1)^𝑖+𝑗𝟙_{{𝑖≥𝑗}}𝑗+𝑁!(𝑖+𝛼−𝑁)_𝑖+1𝑗!𝑖−𝑗!(𝑖+𝛼)_𝑗+𝑁+1^{0≤𝑖,𝑗≤𝑁−1}. (A.61) The generic term of this matrix is ^𝑁−1_𝑘=0𝟙_{{𝑖≥𝑘}}𝑘+𝛼(𝑖)𝑖+𝑁^𝑘(𝑘+𝛼−𝑁)_𝑘×(−1)^𝑗+𝑘𝟙_{{𝑘≥𝑗}}𝑗+𝑁!(𝑘+𝛼−𝑁)_𝑘+1𝑗!𝑘−𝑗!(𝑘+𝛼)_𝑗+𝑁+1=𝟙_{{𝑖≥𝑗}}𝑖!𝑗+𝑁!(𝛼−𝑁)!()!()!×𝑗!𝑖+𝑁𝛼−𝑁−1_{𝑖𝑘=𝑗}(−1)^𝑗+𝑘𝑘+𝛼−𝑗−𝑁−1!()!!()!.𝑖−𝑘𝑘−𝑗𝑘+𝛼−𝑖−𝑁 (A.62) The last sum can be computed as follows: clearly, it vanishes when 𝑖<𝑗 and it equals 1 when 𝑖=𝑗. If 𝑖>𝑗, by using Lemma A.4, ^𝑖_𝑘=𝑗(−1)^𝑗+𝑘𝑘+𝛼−𝑗−𝑁−1!()!!()!=1𝑖−𝑘𝑘−𝑗𝑘+𝛼−𝑖−𝑁𝑖−𝑗!^𝑖_𝑘=𝑗(−1)^𝑗+𝑘𝑖−𝑗!𝑘−𝑗𝑘+𝛼−𝑗−𝑁−1()!=1𝑘+𝛼−𝑖−𝑁𝑖−𝑗!^𝑖−𝑗_𝑘=0(−1)^𝑘𝑘)𝑖−𝑗(𝑘+𝛼−𝑁−1_{𝑖−𝑗−1}=0. (A.63) As a consequence, the entries of the product (A.61) are 𝛿_𝑖𝑗 which proves that the second factor of (A.61) coincides with 𝐋⁻¹.

Lemma A.7. The matrix 𝐋⁻¹𝐁 is given by 𝐋⁻¹()𝐁=−1^𝑖𝛽_𝑁^{𝑖+𝛼−𝛽−1}_{𝛼−𝛽−1𝑁𝑖+𝛼0≤𝑖≤𝑁−1}. (A.64)

Proof. The generic term of 𝐋⁻¹𝐁 is ^𝑁−1_𝑗=0(−1)^𝑖+𝑗𝟙_{{𝑖≥𝑗}}𝑗+𝑁!(𝑖+𝛼−𝑁)_𝑖+1𝑗!𝑖−𝑗!(𝑖+𝛼)_𝑗+𝑁+1𝛽=()!𝑗+𝑁𝛽!𝑖+𝛼−𝑁()!()!×𝑖+𝛼𝛼−𝑁−1_𝑖𝑗=0(−1)^𝑖+𝑗𝑖−𝑗+𝛼−𝑁−1!!!=()!!𝑗!𝑖−𝑗𝛽−𝑗−𝑁𝛽!𝑖+𝛼−𝑁𝑖+𝛼−𝛽−1()!()!×𝑖!𝑖+𝛼𝛼−𝑁−1_𝑖𝑗=0(−1)^𝑖+𝑗𝑖𝑗.𝑖−𝑗+𝛼−𝑁−1𝑖+𝛼−𝛽−1 (A.65) By performing the change of index 𝑗↦𝑖−𝑗 and by using Lemma A.4, the sum in (A.65) is equal to ^𝑖_𝑗=0(−1)^𝑗𝑖𝑗=()𝑗+𝛼−𝑁−1𝑖+𝛼−𝛽−1−1^𝑖=()𝛼−𝑁−1𝛽−𝑁−1^𝑖(𝛼−𝑁−1)!!!.𝛼−𝛽−1𝛽−𝑁 (A.66) By putting this into (A.65), we see that the generic term of 𝐋⁻¹𝐁 writes (−1)^𝑖𝛽!(𝑖+𝛼−𝑁)!𝑖+𝛼−𝛽−1!()!!!=()𝑖!𝑖+𝛼𝛼−𝛽−1𝛽−𝑁−1^𝑖𝛽_𝑁^{𝑖+𝛼−𝛽−1}_{𝛼−𝛽−1𝑁𝑖+𝛼} (A.67) which ends up the proof of Lemma A.7.

Theorem A.8. The matrix 𝐀⁻¹𝐁 is given by 𝐀⁻¹𝐁=(−1)^𝑖𝑁𝑖+𝛼−𝛽^{𝛽𝑁𝑁𝛼−𝛽+𝑁−1𝑖𝑁−1}_𝑁^𝑖+𝛼_{0≤𝑖≤𝑁−1}. (A.68)

Proof. Referring to Lemmas A.5 and A.7, we have that 𝐀⁻¹𝐁=𝐔𝐋⁻¹𝐁=(−1)^𝑖+𝑗𝟙_{{𝑖≤𝑗}}^𝑗_𝑖𝑁^𝑗+𝛼_{𝑁𝑖+𝛼0≤𝑖,𝑗≤𝑁−1}×(−1)^𝑖𝛽_𝑁^{𝑖+𝛼−𝛽−1}_{𝛼−𝛽−1𝑁𝑖+𝛼0≤𝑖≤𝑁−1}. (A.69) The generic term of 𝐀⁻¹𝐁 is (−1)^𝑖𝛽_{𝑁𝑁𝑖+𝛼}^𝑁−1_𝑗=𝑖𝑗𝑖.𝑗+𝛼−𝛽−1𝛼−𝛽−1 (A.70) The foregoing sum can be easily evaluated as follows: ^𝑁−1_𝑗=𝑖𝑗𝑖=𝑗+𝛼−𝛽−1𝛼−𝛽−1𝑖+𝛼−𝛽−1𝛼−𝛽−1^{𝑁−1𝑗=𝑖}=×𝑗+𝛼−𝛽−1𝑖+𝛼−𝛽−1𝑖+𝛼−𝛽−1𝛼−𝛽−1_{𝑁−1𝑗=𝑖}−==𝑁𝑗+𝛼−𝛽𝑖+𝛼−𝛽𝑗+𝛼−𝛽−1𝑖+𝛼−𝛽𝑖+𝛼−𝛽−1𝛼−𝛽−1𝛼−𝛽+𝑁−1𝑖+𝛼−𝛽𝑁𝑖.𝑖+𝛼−𝛽𝛼−𝛽+𝑁−1𝑁−1 (A.71)
Putting this into (A.70) yields the matrix 𝐀⁻¹𝐁 displayed in Theorem A.8.

Table 2 

List of notations.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.