A Newton Interpolation Approach to Generalized Stirling Numbers

Abstract

We employ the generalized factorials to define a Stirling-type pair {𝑠(𝑛,𝑘;𝜶,𝜷,𝑟),𝑆(𝑛,𝑘;𝜶,𝜷,𝑟)} which unifies various Stirling-type numbers investigated by previous authors. We make use of the Newton interpolation and divided differences to obtain some basic properties of the generalized Stirling numbers including the recurrence relation, explicit expression, and generating function. The generalizations of the well-known Dobinski's formula are further investigated.

1. Introduction

Throughout this paper the following notations will be used. We denote by ℝ the set of real numbers and by ℂ the set of complex numbers. Let 𝜶=(𝛼₀,𝛼₁,…) be a vector. If 𝛼_𝑖=𝑖𝜃(𝑖=0,1,…), we denote the vector by 𝜽. We further denote (0,1,…) by 𝟏 and (0,0,…) by 𝟎. Moreover, let us denote the generalized 𝑘th falling factorial of 𝑥 with increment 𝜃 by (𝑥)^(𝑘,𝜃)=𝑥(𝑥−𝜃)⋯(𝑥−𝑘𝜃+𝜃). Particularly, if 𝜃=1, we write (𝑥)^(𝑘)=𝑥(𝑥−1)⋯(𝑥−𝑘+1).

In mathematics, Stirling numbers of the first and second kind, which are named after James Stirling, arise in a variety of combinatorics problems. They have played important roles in combinatorics. Stirling numbers of the first kind are the coefficients in the expansion (𝑥)^(𝑛)=∑^𝑛_𝑘=0𝑠(𝑛,𝑘)𝑥^𝑘, and Stirling numbers of the second kind are characterized by 𝑥^𝑛=∑^𝑛_𝑘=0𝑆(𝑛,𝑘)(𝑥)^(𝑘).

Over the past few decades, there has been an interest in generalizing and extending the Stirling numbers in mathematics literature. By starting with transformations between generalized factorial involving three arbitrary parameters 𝛼,𝛽, and 𝑟, Hsu and Shiue [1] introduced the generalized numbers 𝑆(𝑛,𝑘;𝛼,𝛽,𝑟) and unified those generalizations of the Stirling numbers due to Riordan [2], Carlitz [3, 4], Howard [5], Charalambides-Koutras [6], Gould-Hopper [7], Tsylova [8], and others. They define a Stirling-type pair {𝑆(𝑛,𝑘;𝛼,𝛽,𝑟),𝑠(𝑛,𝑘;𝛼,𝛽,𝑟)} by(𝑥)^(𝑛,𝛼)=^𝑛_𝑘=0𝑆𝑛,𝑘;𝛼,𝛽,𝑟(𝑥−𝑟)^(𝑘,𝛽),(𝑥)^(𝑛,𝛽)=^𝑛_𝑘=0𝑠𝑛,𝑘;𝛼,𝛽,𝑟(𝑥+𝑟)^(𝑘,𝛼). (1.1) They systematically investigated many basic properties including orthogonality relations, recurrence relations, generating function, and the Dobinski identity for their Stirling numbers. Recently, Comtet [9] defines 𝑠_𝛼(𝑛,𝑘) and 𝑆_𝛼(𝑛,𝑘), the generalized Stirling numbers of the first kind and second kind associate with 𝛼₀,𝛼₁,…,𝛼_𝑛−1, by𝑥−𝛼₀𝑥−𝛼₁⋯𝑥−𝛼_𝑛−1=^𝑛_𝑘=0𝑠_𝛼(𝑛,𝑘)𝑥^𝑘,𝑥^𝑛=^𝑛_𝑘=0𝑆_𝛼(𝑛,𝑘)𝑥−𝛼₀𝑥−𝛼₁⋯𝑥−𝛼_𝑘−1. (1.2) El-Desouky [10] modified the noncentral Stirling numbers of the first and second kind. He defined the multiparameter noncentral Stirling numbers of the first kind and second kind as follows:𝑥−𝛼₀𝑥−𝛼₁⋯𝑥−𝛼_𝑛−1=^𝑛_𝑘=0𝑆(𝑛,𝑘;𝛼)(𝑥)^(𝑘),(𝑥)^(𝑛)=^𝑛_𝑘=0𝑠(𝑛,𝑘;𝛼)𝑥−𝛼₀𝑥−𝛼₁⋯𝑥−𝛼_𝑘−1. (1.3)(1.4) The recurrence relations, generating functions, and explicit forms for El-Desouky's Stirling numbers are obtained.

In another direction, Stirling numbers and their generalizations were investigated via differential operators. Carlitz and Klamkin [11] defined the Stirling numbers of the second kind by(𝑥𝐷)^𝑛=^𝑛_𝑘=1𝑆(𝑛,𝑘)𝑥^𝑘𝐷^𝑘, (1.5) where 𝐷 is a differential operator 𝑑/𝑑𝑥. Actually, this can be traced back at least to Scherk [12]. In the physical literature, Katriel [13] discovered (1.5) was in connection with the normal ordering expressions in the boson creation operator 𝑎^† and annihilation 𝑎, satisfying the commutation relation [𝑎,𝑎^†]=1 of the Weyl algebra. Recently, Lang [14, 15] generalized the stirling numbers of the second kind by the following operator identity:𝑥^𝑟𝐷^𝑛=^𝑛_𝑘=1𝑆(𝑟;𝑛,𝑘)𝑥^𝑘𝐷^𝑘, (1.6) where 𝑟 is a nonnegative integer. He further obtained many properties of these numbers. More recently, Blasiak et al. [16] defined 𝑆_𝑟,𝑠(𝑛,𝑘), the generalized Stirling numbers of the second kind arising in the solution of the general normal ordering problem for a boson string, as follows𝑥^𝑟𝐷^𝑠𝑛=𝑥^{𝑛(𝑟−𝑠)𝑛𝑠}_𝑘=𝑠𝑆_𝑟,𝑠(𝑛,𝑘)𝑥^𝑘𝐷^𝑘. (1.7) These numbers were firstly defined by Carlitz [17]. More generally, given two sequences of nonnegative integers 𝐫=(𝑟₁,𝑟₂,…,𝑟_𝑛) and 𝐬=(𝑠₁,𝑠₂,…,𝑠_𝑛), Blasiak [18] generalized this formula by𝑥^𝑟𝑛𝐷^𝑠𝑛⋯𝑥^𝑟2𝐷^𝑠2𝑥^𝑟1𝐷^𝑠1=𝑥^{𝑑𝑛𝑠1+𝑠2+⋯+𝑠𝑛}_𝑘=𝑠1𝑆_𝐫,𝐬(𝑛,𝑘)𝑥^𝑘𝐷^𝑘, (1.8) where 𝑑_𝑛=∑^𝑛_𝑘=1(𝑟_𝑘−𝑠_𝑘). He gave an explicit formula for the generalized Stirling numbers 𝑆_𝐫,𝐬(𝑛,𝑘). In [19], a different explicit expression for these numbers was presented.

By considering powers (𝑉𝑈)^𝑛 of the noncommuting variables 𝑈, 𝑉 satisfying 𝑈𝑉=𝑉𝑈+ℎ𝑉^𝑠, Mansour and Schork [20] introduced a new family of generalized Stirling numbers 𝔖_𝑠;ℎ(𝑛,𝑘) as(𝑉𝑈)^𝑛=^𝑛_𝑘=1𝔖_𝑠;ℎ(𝑛,𝑘)𝑉^{𝑠(𝑛−𝑘)+𝑘}𝑈^𝑘, (1.9) which reduced to the conventional Stirling numbers of second kind and Bell numbers in the case 𝑠=0,ℎ=1. As mentioned in [21], this type of generalized Stirling numbers is not a special case of Howard's degenerate weight Stirling numbers although they look very similar.

Moreover, for any sequence of real numbers 𝜶=(𝛼₀,𝛼₁,…,𝛼_𝑛−1) and a sequence of nonnegative integers 𝐫=(𝑟₀,𝑟₁,…,𝑟_𝑛−1), by using operational identity [22, 23] El-Desouky and Cakić [24] defined a generalized multiparameter noncentral Stirling numbers of the second kind 𝑆(𝑛,𝑘;𝜶,𝐫) by^𝑛−1_𝑗=0𝑥^𝛼𝑖𝛿𝑥^{−𝛼𝑗𝑟𝑗}=^𝑛−1_𝑗=0𝛿−𝛼_𝑗^𝑟𝑗=^|𝐫|_𝑘=0𝑆(𝑛,𝑘;𝜶,𝐫)𝑥^𝑘𝐷^𝑘, (1.10) where |𝐫|=𝑟₀+𝑟₁+⋯+𝑟_𝑛−1. These numbers reduced to the multiparameter noncentral Stirling numbers of the second kind 𝑆(𝑛,𝑘;𝛼) in (1.3) if all 𝑟_𝑖=1.

As a useful tool, the Newton interpolation with divided differences was utilized to obtain closed formulas for Dickson-Stirling numbers in the paper [25] provided by the referee. In this paper, we make use of the generalized factorials to define a Stirling-type pair {𝑠(𝑛,𝑘;𝜶,𝜷,𝑟),𝑆(𝑛,𝑘;𝜶,𝜷,𝑟)} which unifies various Stirling-type numbers investigated by previous authors. By using the Newton interpolation and divided differences, we obtain the basic properties including recurrence relations, explicit expression, and generating function. The generalizations of the well-known Dobinski's formula are further investigated. This paper is organized as follows. In Section 2, we introduce the Newton interpolation and divided differences. Several important properties of divided differences are presented. In Section 3, the definitions of a new family of Stirling numbers are given. According to the definitions, the recurrence relation as well as an explicit formula is derived. Moreover, we also investigate the generating function for our generalized Stirling numbers. In views of our results, we rediscover many interesting special cases which are introduced in the above. Finally, in Section 4, the associated generalized Bell numbers and Bell polynomials are presented. Furthermore, a generalized Dobinski's formula is derived.

2. Divided Differences and Newton Interpolation

For a sequence of points 𝜶=(𝛼₀,𝛼₁,…) and all 𝛼_𝑖∈ℝ or ℂ, we define𝜔_0,𝜶(𝑥)=1,𝜔_𝑛,𝜶(𝑥)=^𝑛−1_𝑖=0𝑥−𝛼_𝑖,𝑛=1,2,…. (2.1) Let 𝑁_𝑛(𝑥) be the Newton interpolating polynomial of degree at most 𝑛 that interpolates a function 𝑓(𝑥) at the point 𝛼₀,𝛼₁,…,𝛼_𝑛; then this polynomial is given as in𝑁_𝑛(𝑥)=^𝑛_𝑖=0Δ𝛼₀,…,𝛼_𝑖𝑓⋅𝜔_𝑖,𝜶(𝑥), (2.2) where Δ(𝛼₀,…,𝛼_𝑖)𝑓 is the divided difference of the 𝑖th order of the function 𝑓. As is well known, for the distinct points 𝛼₀,𝛼₁,…,𝛼_𝑛, the divided differences of the function 𝑓 are defined recursively by the following formula:Δ𝛼₀𝑓=𝑓𝛼₀,Δ𝛼₀,…,𝛼_𝑛Δ𝛼𝑓=⁰,…,𝛼_𝑛−1𝑓−Δ𝛼₁,…,𝛼_𝑛𝑓𝛼₀−𝛼_𝑛. (2.3)(2.4)

Divided differences as the coefficients of the Newton interpolating polynomial have played an important role in numerical analysis, especially in interpolation and approximation by polynomials and in spline theory; see [26] for a recent survey. They also have many applications in combinatorics [27, 28]. The divided differences can be expressed by the explicit formulaΔ𝛼₀,…,𝛼_𝑛𝑓=^𝑛_𝑖=0𝑓𝛼_𝑖∏^𝑛_{𝑗=0,≠𝑖}𝛼_𝑖−𝛼_𝑗. (2.5) From the above expression it is not difficult to find the divided differences are symmetric functions of their arguments. In particular, taking 𝛼_𝑖=𝛼+𝑖𝜃(𝜃≠0) we haveΔ()1𝛼,…,𝛼+𝑛𝜃𝑓=𝑛!𝜃^𝑛Δ^𝑛_𝜃𝑓(𝛼)=1𝑛!𝜃^𝑛𝑛_𝑖=0𝑛𝑖(−1)^𝑛−𝑖𝑓(𝛼+𝑖𝜃), (2.6) where Δ_𝜃 is the difference operator with step size 𝜃.

Divided differences can be extended to the cases with repeated points. From the recursive formula (2.4), it is clear that if 𝛼₀≠𝛼₁ the following holds:Δ𝛼₀,𝛼₁𝑓𝛼𝑓=¹−𝑓𝛼₀𝛼₁−𝛼₀. (2.7) If repetitions are permitted in the arguments and the function 𝑓 is smooth enough, thenlim_{𝛼1→𝛼0}Δ𝛼₀,𝛼₁𝑓=lim_{𝛼1→𝛼0}𝑓𝛼₁−𝑓𝛼₀𝛼₁−𝛼₀=𝑓^′𝛼₀. (2.8) This gives the definition of first-order divided differences with repeated pointsΔ𝛼₀,𝛼₀𝑓=𝑓^′𝛼₀. (2.9) In general, let 𝛼₀≤𝛼₁≤⋯≤𝛼_𝑛. Then the divided differences with repeated points obey the following recursive formula:Δ𝛼₀,…,𝛼_𝑛Δ𝛼𝑓=⁰,…,𝛼_𝑛−1𝑓−Δ𝛼₁,…,𝛼_𝑛𝑓𝛼₀−𝛼_𝑛,if𝛼_𝑛≠𝛼₀,𝑓^(𝑛)𝛼₀𝑛!,if𝛼_𝑛=𝛼₀. (2.10) It is evident that divided differences can be viewed as a discrete analogue of derivatives. If 𝛼_𝑛=𝛼₀, then all the points 𝛼₀,𝛼₁,…,𝛼_𝑛 are the same. In this case, 𝑁_𝑛(𝑥) in (2.2) is the Taylor polynomial of the function 𝑓 at the point 𝛼₀. More generally, if {𝛼₀,𝛼₁,…,𝛼_𝑛}={𝛼^′₀,…,𝛼^′_0𝑝0,𝛼^′₁,…,𝛼^′_1𝑝1,…,𝛼^′_𝑚,…,𝛼^′_𝑚𝑝𝑚} and 𝑝₀+𝑝₁+⋯+𝑝_𝑚=𝑛+1 where 𝛼^′₀,𝛼^′₁,…,𝛼^′_𝑚 are distinct, we defineΩ_𝑖(𝑥)=^𝑚_{𝑘=0,≠𝑖}𝑥−𝛼^′_𝑘^𝑝𝑘,𝑆_𝑙𝑖(𝑥)=^𝑚_{𝑘=0,≠𝑖}𝑝_𝑘𝛼^′_𝑘−𝑥^𝑙, (2.11) with 𝑙≥1,0≤𝑖≤𝑚. Recall that the cycle index of symmetric group𝑍_𝑛𝑥_𝑘=𝑍_𝑛𝑥₁,𝑥₂,…,𝑥_𝑛=_{𝑎1+2𝑎2+⋯+𝑛𝑎𝑛=𝑛}1𝑎₁!(1)^𝑎1𝑎₂!(2)^𝑎2⋯𝑎_𝑛!(𝑛)^𝑎𝑛𝑥^𝑎1₁𝑥^𝑎2₂⋯𝑥^𝑎𝑛_𝑛 (2.12) is one of the essential tools in enumerative combinatorics [29]. Using the cycle index of symmetric group, the divided differences with repeated points can be expressed by the following explicit formula [30] (see also [31]):Δ𝛼₀,…,𝛼_𝑛𝑓=^𝑚_𝑖=0Ω_𝑖𝛼^′_𝑖^{𝑝−1𝑖−1}_𝑗=0𝑍_{𝑝𝑖−1−𝑗}𝑆_!𝑖𝛼^′_𝑖𝑓^(𝑗)𝛼^′_𝑖,𝑗! (2.13) where𝑍_{𝑝𝑖−1−𝑗}𝑆_!𝑖𝛼^′_𝑖=𝑍_{𝑝𝑖−1−𝑗}𝑆_1𝑖𝛼^′_𝑖,𝑆_2𝑖𝛼^′_𝑖,…,𝑆_{𝑝𝑖−1−𝑗𝑖}𝛼^′_𝑖. (2.14)

It is well known that the Leibniz formula for higher derivatives is basic and important in calculus. A divided difference form of this formula given by [32] is stated as below. Let ℎ=𝑓𝑔. If 𝑓 and 𝑔 are sufficiently smooth functions, then for arbitrary points 𝛼₀,𝛼₁,…,𝛼_𝑛,Δ𝛼₀,…,𝛼_𝑛ℎ=^𝑛_𝑖=0Δ𝛼₀,…,𝛼_𝑖𝑓⋅Δ𝛼_𝑖,…,𝛼_𝑛𝑔. (2.15) This formula is called the Steffensen formula which is a generalization of the Leibniz formula. If 𝛼₀=𝛼₁=⋯=𝛼_𝑛, then the Leibniz formula holds, namely,ℎ^(𝑛)𝛼₀=^𝑛_𝑖=0𝑛𝑖𝑓^(𝑖)𝛼₀𝑔^{(𝑛−𝑖)}𝛼₀. (2.16)

3. Generalized Stirling Numbers

Let 𝜶=(𝛼₀,𝛼₁,…) and 𝜷=(𝛽₀,𝛽₁,…) be two vectors. We define two kinds of Stirling-type numbers as𝜔_𝑛,𝜶(𝑥)=^𝑛_𝑘=0𝑆(𝑛,𝑘;𝜶,𝜷,𝑟)𝜔_𝑘,𝜷(),𝜔𝑥−𝑟_𝑛,𝜷(𝑥)=^𝑛_𝑘=0𝑠(𝑛,𝑘;𝜶,𝜷,𝑟)𝜔_𝑘,𝜶(𝑥+𝑟), (3.1)(3.2) where 𝑆(𝑛,𝑘;𝜶,𝜷,𝑟) are called the generalized Stirling numbers of the second kind with the parameters 𝜶, 𝜷, and 𝑟, and 𝑠(𝑛,𝑘;𝜶,𝜷,𝑟) are called the generalized Stirling numbers of the first kind. It is obvious that 𝑆(𝑛,𝑘;𝜶,𝜷,𝑟)=𝑠(𝑛,𝑘;𝜷,𝜶,−𝑟). In particular, 𝑆(𝑛,𝑘;𝟎,𝟏,0) is the conventional Stirling number of the second kind, and 𝑠(𝑛,𝑘;𝟎,𝟏,0) is of the first kind.

In this section, making use of divided difference operator and the Newton interpolation in Section 2, we will investigate orthogonality relations, recurrences, explicit expressions, and generating functions for the generalized Stirling numbers.

3.1. Basic Properties of the Generalized Stirling Numbers

Firstly, let us consider orthogonality relations of the two kinds of the generalized Stirling numbers. By substituting (3.1) into (3.2) and (3.2) into (3.1), one may easily get the following orthogonality relations ^𝑛_𝑖=𝑘𝑠(𝑛,𝑖;𝜶,𝜷,𝑟)𝑆(𝑖,𝑘;𝜶,𝜷,𝑟)=𝛿_𝑛,𝑘,_{𝑛𝑖=𝑘}𝑆(𝑛,𝑖;𝜶,𝜷,𝑟)𝑠(𝑖,𝑘;𝜶,𝜷,𝑟)=𝛿_𝑛,𝑘, (3.3)(3.4) respectively, where the Kronecker symbol 𝛿_𝑛,𝑘 is defined by 𝛿_𝑛,𝑘=1 if 𝑛=𝑘, and 𝛿_𝑛,𝑘=0 if 𝑛≠𝑘. As a consequence, the inverse relations are immediately obtained:𝑓_𝑛=^𝑛_𝑘=0𝑆(𝑛,𝑘;𝜶,𝜷,𝑟)𝑔_𝑘⟺𝑔_𝑛=^𝑛_𝑘=0𝑠(𝑛,𝑘;𝜶,𝜷,𝑟)𝑓_𝑘. (3.5)

Next, from the definition (3.1), one may see that 𝑆(𝑛,𝑘;𝜶,𝜷,𝑟) can be viewed as the coefficients of the Newton interpolation of the function 𝜔_𝑛,𝜶 at the points 𝑟+𝛽₀,𝑟+𝛽₁,…,𝑟+𝛽_𝑛. Thus, we immediately have the following theorem.

Theorem 3.1. For arbitrary parameters 𝜶, 𝜷, and 𝑟, there holds 𝑆(𝑛,𝑘;𝜶,𝜷,𝑟)=Δ𝑟+𝛽₀,…,𝑟+𝛽_𝑘𝜔_𝑛,𝜶. (3.6) In particular, if 𝛽₀,𝛽₁,…,𝛽_𝑛 are distinct, we have 𝑆(𝑛,𝑘;𝜶,𝜷,𝑟)=^𝑘_𝑖=0∏^𝑛−1_𝑗=0𝑟+𝛽_𝑖−𝛼_𝑗∏^𝑘_{𝑗=0,≠𝑖}𝛽_𝑖−𝛽_𝑗. (3.7) If 𝛽₀=𝛽₁=⋯=𝛽_𝑛=0, then 𝑆()=1𝑛,𝑘;𝜶,𝟎,𝑟𝑘!_{0≤𝑖1<⋯<𝑖𝑛−𝑘≤𝑛−1}𝑟−𝛼_𝑖1⋯𝑟−𝛼_{𝑖𝑛−𝑘}. (3.8)

This theorem gives the explicit expressions for the generalized Stirling numbers. We can similarly get 𝑠(𝑛,𝑘;𝜶,𝜷,𝑟)=Δ(−𝑟+𝛼₀,…,−𝑟+𝛼_𝑘)𝜔_𝑛,𝜷. By (3.6), we can further get the recurrence relations as follows.

Theorem 3.2. For arbitrary parameters 𝜶, 𝜷, and 𝑟, there holds 𝑆(𝑛,𝑘;𝜶,𝜷,𝑟)=𝑆(𝑛−1,𝑘−1;𝜶,𝜷,𝑟)+𝑟+𝛽_𝑘−𝛼_𝑛−1𝑆(𝑛−1,𝑘;𝜶,𝜷,𝑟). (3.9) In particular, we have 𝑆(𝑛,0;𝜶,𝜷,𝑟)=𝜔_𝑛,𝜶𝑟+𝛽₀=𝑟+𝛽₀−𝛼₀⋯𝑟+𝛽₀−𝛼_𝑛−1. (3.10)

Proof. According to (3.6), we have 𝑆(𝑛,𝑘;𝜶,𝜷,𝑟)=Δ𝑟+𝛽₀,…,𝑟+𝛽_𝑘𝜔_{𝑛−1,𝜶}⋅−𝛼_𝑛−1. (3.11) By using the Steffensen formula for divided differences and the basic facts Δ𝑥₀,…,𝑥_𝑖𝜔_𝑗,𝜶=𝛿_𝑖,𝑗, (3.12) we have 𝑆(𝑛,𝑘;𝜶,𝜷,𝑟)=Δ𝑟+𝛽₀,…,𝑟+𝛽_𝑘−1𝜔_{𝑛−1,𝜶}+𝑟+𝛽_𝑘−𝛼_𝑛−1Δ𝑟+𝛽₀,…,𝑟+𝛽_𝑘𝜔_{𝑛−1,𝜶}. (3.13) This leads to (3.9), and the proof is complete.

Finally, let us consider the generating function of the Stirling numbers 𝑆(𝑛,𝑘;𝜶,𝜷,𝑟) denoted by 𝐺(𝑡;𝑘,𝜶,𝜷,𝑟). Assume that 𝐺(𝑡;𝑘,𝜶,𝜷,𝑟) is of the form:𝐺(𝑡;𝑘,𝜶,𝜷,𝑟)=^∞_𝑛=0𝐴_𝑛𝑆(𝑛,𝑘;𝜶,𝜷,𝑟)𝑡^𝑛, (3.14) where 𝐴₀,𝐴₁,… is a reference sequence. In this way we treat at the same time the case of ordinary coefficients of 𝐺(𝐴_𝑛=1) and the case of Taylor coefficients (𝐴_𝑛=1/𝑛!). Let Φ(𝑥,𝑡)=∑^∞_𝑛=0𝐴_𝑛𝜔_𝑛,𝜶(𝑥)𝑡^𝑛. Making use of (3.6), we get the following:𝐺(𝑡;𝑘,𝜶,𝜷,𝑟)=^∞_𝑛=0𝐴_𝑛Δ𝑟+𝛽₀,…,𝑟+𝛽_𝑘𝜔_𝑛,𝜶𝑡^𝑛=Δ𝑟+𝛽₀,…,𝑟+𝛽_𝑘Φ(⋅,𝑡). (3.15) This formula is essential and important for getting the generating function of the generalized Stirling numbers. If we get the analytic expression of Φ(𝑥,𝑡) by choosing special 𝜶, the analytic expression of 𝐺(𝑡;𝑘,𝜶,𝜷,𝑟) is obtained as well.

3.2. Special Cases

Because the parameters 𝜶,𝜷, and 𝑟 are arbitrary, our results contain many interesting special cases. In this part we will investigate these special cases. Some results have been derived and some are new.

Let 𝜽=(0,𝜃,…) and all 𝛽_𝑖 be distinct. According to Theorems 3.1 and 3.2, we have the explicit expressions and the recurrence relations for new generalized Stirling numbers 𝑆(𝑛,𝑘;𝜽,𝜷,𝑟).

Corollary 3.3. The numbers 𝑆(𝑛,𝑘;𝜽,𝜷,𝑟) have the following explicit expression 𝑆𝑛,𝑘;𝜽,𝜷,𝑟=^𝑘_𝑖=0∏^𝑛−1_𝑗=0𝑟+𝛽_𝑖−𝑗𝜃∏^𝑘_{𝑗=0,≠𝑖}𝛽_𝑖−𝛽_𝑗. (3.16)

Corollary 3.4. The numbers 𝑆(𝑛,𝑘;𝜽,𝜷,𝑟) satisfy the following recurrence relation 𝑆𝑛,𝑘;𝜽,𝜷,𝑟=𝑆𝑛−1,𝑘−1;𝜽,𝜷,𝑟+𝑟+𝛽_𝑘−(𝑛−1)𝜃𝑆𝑛−1,𝑘;𝜽,𝜷,𝑟. (3.17)

For 𝜽=(0,𝜃,…) and 𝐴_𝑛=1/𝑛!, if 𝜃≠0 we haveΦ(𝑥,𝑡)=^∞_𝑛=0(𝑥)^(𝑛,𝜃)𝑡^𝑛=()𝑛!1+𝜃𝑡^𝑥/𝜃, (3.18) and if 𝜃=0 we haveΦ(𝑥,𝑡)=^∞_𝑛=0𝑥^𝑛𝑡^𝑛𝑛!=𝑒^𝑥𝑡. (3.19) Thus, by (3.15) one easily obtain the following theorem.

Theorem 3.5. The sequence {𝑆(𝑛,𝑘;𝜽,𝜷,𝑟)} has the following exponential generating function: ^∞_𝑛=0𝑆𝑛,𝑘;𝑡𝜽,𝜷,𝑟^𝑛=()𝑛!1+𝜃𝑡^{𝑘𝑟/𝜃}_𝑖=0(1+𝜃𝑡)^{𝛽𝑖/𝜃}∏^𝑘_{𝑗=0,≠𝑖}𝛽_𝑖−𝛽_𝑗𝑒,𝜃≠0,_{𝑘𝑟𝑡𝑖=0}𝑒^𝛽𝑖𝑡∏^𝑘_{𝑗=0,≠𝑖}𝛽_𝑖−𝛽_𝑗,𝜃=0. (3.20)

Our generalized Stirling numbers 𝑆(𝑛,𝑘;𝜽,𝜷,𝑟) include the Stirling numbers due to Hsu and Shiue [1], EI-Desouky's multiparameter noncentral Stirling numbers [10], and the so-called Comtet numbers [9] as special cases. Now, let us discuss these special cases as follows.

Example 3.6. Let 𝜷=𝜽^′∶=(0,𝜃^′,…) and 𝜃^′≠0. This implies the points 𝛽_𝑖 are equally spaced with step size 𝜃^′. By Corollaries 3.3 and 3.4, we immediately get the explicit expression for the generalized Stirling numbers 𝑆𝑛,𝑘;𝜽,𝜽^′=1,𝑟𝑘!𝜃^𝑘′𝑘_𝑖=0(−1)^𝑘−𝑖𝑘𝑖^{𝑛−1𝑗=0}𝑟+𝑗𝜃′−𝑖𝜃 (3.21) and the recurrence relation 𝑆𝑛+1,𝑘;𝜽,𝜽^′,𝑟=𝑆𝑛,𝑘−1;𝜽,𝜽^′,𝑟+𝑟+𝑘𝜃^′−𝑛𝜃𝑆𝑛,𝑘;𝜽,𝜽^′,𝑟. (3.22) For 𝜃≠0, the following holds ^𝑘_𝑖=0(1+𝜃𝑡)^{(𝑟+𝑖𝜃′)/𝜃}∏^𝑘_{𝑗=0,≠𝑖}𝑖𝜃^′−𝑗𝜃^′=1𝑘!𝜃^′𝑘(1+𝜃𝑡)^{𝑘𝑟/𝜃}_𝑖=0(−1)^𝑘−𝑖𝑘𝑖(1+𝜃𝑡)^{𝑖𝜃′/𝜃}=1𝑘!𝜃^′𝑘(1+𝜃𝑡)^𝑟/𝜃(1+𝜃𝑡)^𝜃′/𝜃−1^𝑘. (3.23) In a similar manner, we can also get the generating function for 𝜃=0. Thus, we have ^∞_𝑛=0𝑆𝑛,𝑘;𝜽,𝜽^′𝑡,𝑟^𝑛=1𝑛!()𝑘!1+𝜃𝑡^𝑟/𝜃(1+𝜃𝑡)^𝜃′/𝜃−1𝜃^′𝑘1,𝜃≠0,𝑒𝑘!^𝑟𝑡𝑒^𝜃′𝑡−1𝜃^′𝑘,𝜃=0. (3.24)
Here 𝑆(𝑛,𝑘;𝜽,𝜽′,𝑟) is equivalent to 𝑆(𝑛,𝑘;𝛼,𝛽,𝑟) in [1]. As mentioned in [1], the generalized Stirling numbers 𝑆(𝑛,𝑘;𝜽,𝜽′,𝑟) contain serval special cases, for example, two kinds of the classical Stirling numbers, the binomial coefficients, the Lah numbers, Carlitz's two kinds of weighted Stirling numbers [4], Carlitz's two kinds of degenerate Stirling numbers [3], Howard's weighted degenerate Stirling numbers [5], Gould-Hopper's noncentral Lah numbers [7], Riordan's noncentral Stirling numbers [2], the noncentral 𝐶 numbers extensively studied by Charalambides and Koutras [6], Tsylova's numbers [8], Todorov's numbers [33], Nandi and Dutta's associated Lah numbers [34], and the 𝑟-Stirling numbers of the first kind fully developed by Broder [35]. Hsu and Shiue obtained the recurrence relation for the generalized Stirling numbers 𝑆(𝑛,𝑘;𝜽,𝜽′,𝑟), and they also found the generating function by solving a difference-differential equation. However, the formula (3.21) was new and not given by [1]. Obviously, in the present paper we follow a very different approach to rediscover the recurrence relation and the generating function. In the case 𝑟=0, one may also refer to [36].
It is remarkable that Mansour and Schork [20] recently considered 𝑈𝑉−𝑉𝑈=ℎ𝑉^𝑠 to generalize the commutation relation 𝑈𝑉−𝑉𝑈=1. They defined generalized Stirling numbers 𝔖_𝑠;ℎ(𝑛,𝑘) by (1.9). The explicit expressions of these generalized Stirling numbers are given by [20] (see also [21]), and they are very closely related to the numbers considered by Lang [14]. In [21], the authors exploited many properties of these generalized Stirling numbers. It is interesting that observing our generalized Stirling numbers 𝑆(𝑛,𝑘;𝜽,𝜷,𝑟) by 𝜃=−𝑠ℎ,𝛽_𝑘=𝑘ℎ(1−𝑠) and 𝑟=0, we find that the Stirling numbers due to Mansour and Schork are actually a special case of ours and Hsu-Shiue's, and they are equivalent to the numbers due to [36]. Thus, by (3.20) we get the exponential generating function of the generalized Stirling numbers due to Mansour and Schork: 1()𝑘!1−ℎ𝑠𝑡^{(𝑠−1)/𝑠}−1ℎ(1−𝑠)^𝑘=^∞_𝑛=0𝔖_𝑠;ℎ()𝑡𝑛,𝑘^𝑛𝑛!. (3.25) In [21], the authors gave the generating function for 𝑘=1.

Example 3.7. Let 𝜃=1,𝑟=0, and 𝛽_𝑖 be arbitrary but distinct, and one can get the multiparameter noncentral Stirling numbers of the first kind introduced by El-Desouky [10]. Here we denote the numbers by 𝑆(𝑛,𝑘;𝟏,𝜷,0). Using Corollaries 3.3 and 3.4 and Theorem 3.5, we rediscover the explicit expression, recurrence relation, and the generating function, namely, 𝑆𝑛,𝑘;𝟏,𝜷,0=^𝑘_𝑖=0∏^𝑛−1_𝑗=0𝛽_𝑖−𝑗∏^𝑘_{𝑗=0,≠𝑖}𝛽_𝑖−𝛽_𝑗,𝑆𝑛+1,𝑘;𝟏,𝜷,0=𝑆𝑛,𝑘−1;𝟏,𝜷,0+𝛽_𝑘−𝑛𝑆𝑛,𝑘;𝟏,𝜷,0_∞𝑛=0,𝑆𝑛,𝑘;𝑡𝟏,𝜷,0^𝑛=𝑛!^𝑘_𝑖=0(1+𝑡)^𝛽𝑖∏^𝑘_{𝑗=0,≠𝑖}𝛽_𝑖−𝛽_𝑗. (3.26)

Example 3.8. Let us consider the case 𝜽=𝟎. In this case, there holds 𝑥^𝑛=^𝑛_𝑘=0𝑆(𝑛,𝑘;𝟎,𝜷,𝑟)𝜔_𝑘,𝜷(𝑥−𝑟), (3.27) which is equivalent to (𝑥+𝑟)^𝑛=^𝑛_𝑘=0𝑆(𝑛,𝑘;𝟎,𝜷,𝑟)𝜔_𝑘,𝜷(𝑥). (3.28) Especially, for 𝑟=0 the Comtet numbers [9] (see also [10]) are defined associated with the sequence 𝜷 by 𝑥^𝑛=^𝑛_𝑘=0𝑆_𝜷(𝑛,𝑘)𝑥−𝛽₀𝑥−𝛽₁⋯𝑥−𝛽_𝑘−1. (3.29) This implies 𝑆(𝑛,𝑘;𝟎,𝜷,0)=𝑆_𝜷(𝑛,𝑘). Thus, it is not difficult to obtain 𝑆(𝑛,𝑘;𝟎,𝜷,0)=^𝑘_𝑖=0𝛽^𝑛_𝑖∏^𝑘_{𝑗=0,≠𝑖}𝛽_𝑖−𝛽_𝑗,𝑆(𝑛+1,𝑘;𝟎,𝜷,0)=𝑆(𝑛,𝑘−1;𝟎,𝜷,0)+𝛽_𝑘𝑆(𝑛,𝑘;𝟎,𝜷,0), (3.30) and the exponential generating function ^∞_𝑛=0𝑆()𝑡𝑛,𝑘;𝟎,𝜷,0^𝑛=𝑛!^𝑘_𝑖=0𝑒^𝛽𝑖𝑡∏^𝑘_{𝑗=0,≠𝑖}𝛽_𝑖−𝛽_𝑗. (3.31) It is worth noting that the Comtet numbers 𝑆(𝑛,𝑘;𝟎,𝜷,0) can be rewritten as an alternate form 𝑆(𝑛,𝑘;𝟎,𝜷,0)=_{0≤𝑖1≤⋯≤𝑖𝑛≤𝑘}𝛽_𝑖1𝛽_𝑖2⋯𝛽_𝑖𝑛, (3.32) which is really the complete symmetric function of 𝑛th order with respect to the variables 𝛽₀, 𝛽₁, …, 𝛽_𝑘. By (3.15), they have the ordinary generating function: ^∞_𝑛=0𝑆(𝑛,𝑘;𝟎,𝜷,0)𝑡^𝑛=Δ𝛽₀,…,𝛽_𝑘1(⋅)𝑡=𝑡1−∏^𝑘_𝑖=01^𝑘−𝛽_𝑖𝑡. (3.33)
Moreover, if we let 𝑟=−𝑎,𝜷=𝟏, then we get the noncentral Stirling numbers of the second kind defined by Koutras [37] (see also [24]). For more details one refers to [37].

What has been discussed above in this subsection is relevant to the generalized Stirling numbers with equidistance parameters 𝛼_𝑖. However, we are also interested in the other cases. In a recent year, many authors [14, 16, 18, 19, 24] were devoted to the generalized Stirling numbers by differential operator. We here rediscover these generalized Stirling numbers by the Newton interpolation.

Example 3.9. Our generalized Stirling numbers 𝑆(𝑛,𝑘;𝜶,𝜷,𝑟) also contain the numbers due to Blasiak [18] as a special case. Here we let 𝐫=(𝑟₁,𝑟₂,…,𝑟_𝑚), 𝐬=(𝑠₁,𝑠₂,…,𝑠_𝑚) and let 𝑑₀=0 and 𝑑_𝑚=∑^𝑚_𝑖=1(𝑟_𝑖−𝑠_𝑖) for 𝑚≥1. Moreover, we let 𝐝=−𝑑₀,𝑑₀−1,…,𝑑₀−𝑠₁+1,𝑑₁,𝑑₁−1,…,𝑑₁−𝑠₂𝑑+1,…,_𝑚−1,𝑑_𝑚−1−1,…,𝑑_𝑚−1−𝑠_𝑚,̂+1𝐬=0,1,…,𝑠₁+𝑠₂+⋯+𝑠_𝑚, (3.34) where 𝑠₁+𝑠₂+⋯+𝑠_𝑚=𝑛. By using (3.7) we immediately have the explicit expression of 𝑆(𝑛,𝑘;𝐝,𝐬,̂0) as follows 𝑆̂=1𝑛,𝑘;𝐝,𝐬,0𝑘!^𝑘_𝑗=0𝑘𝑗(−1)^{𝑚𝑘−𝑗}_𝑖=1𝑑_𝑖−1+𝑗^(𝑠𝑖), (3.35) which is in accordance with the generalized Stirling numbers 𝑆_𝐫,𝐬(𝑚,𝑘) introduced by Blasiak [18]. Blasiak got this explicit formula by using the operator 𝑥^𝑟𝑚𝐷^𝑠𝑚⋯𝑥^𝑟2𝐷^𝑠2𝑥^𝑟1𝐷^𝑠1 to act on 𝑒^𝑥. His proof is very different from ours. Recently, El-Desouky et al. [19] found a new expression by successive application of Leibniz formula. The special case 𝐫=(𝑟,𝑟,…,𝑟) and 𝐬=(𝑠,𝑠,…,𝑠) is investigated by Blasiak et al. [16], and they gave us Lang's result [14] as a special case for 𝐬=(1,1,…,1).

Example 3.10. By operating with (1.10) on 𝑒^𝑥 and using Cauchy rule of multiplication of series, El-Desouky and Cakić [24] obtain the explicit formula 𝑆(𝑛,𝑘;𝜶,𝐫)=1𝑘!^𝑘_𝑖=0𝑘𝑖(−1)^{𝑘−𝑖𝑛−1}_𝑗=0𝑘−𝛼_𝑗^𝑟𝑗. (3.36) In fact, let 𝜶={𝛼₀,…,𝛼_0𝑟0,𝛼₁,…,𝛼_1𝑟1,…,𝛼_𝑛−1,…,𝛼_{𝑛−1𝑟𝑛−1}}. It is not difficult to find 𝑆(𝑛,𝑘;𝜶,𝐫)=𝑆(𝑟₀+𝑟₁+⋯+𝑟_𝑛−1,𝑘;𝜶,𝟏,0) holds. In particular, setting 𝑛=2, 𝛼₀=0, 𝛼₁=1, 𝑟₀=𝑙, and 𝑟₁=𝑚−𝑙 in 𝑆(𝑟₀+𝑟₁+⋯+𝑟_𝑛−1,𝑘;𝜶,𝟏,0), we get the explicit expression of the number of partitions of 𝑀={𝑥₁,𝑥₂,…,𝑥_𝑚} into 𝑛 nonempty parts such that the distance of any two members in the same part differs from 𝑙 denoted by 𝑇_𝑙(𝑚,𝑘); see [38].

4. Generalized Bell Polynomials and Dobinski-Type Formulas

Recall that the Bell numbers 𝐵_𝑛 and the exponential polynomials 𝐵_𝑛(𝑥) are defined, respectively, by the sums𝐵_𝑛=^𝑛_𝑘=0𝑆(𝑛,𝑘),𝐵_𝑛(𝑥)=^𝑛_𝑘=0𝑆(𝑛,𝑘)𝑥^𝑘. (4.1) The Bell polynomials 𝐵_𝑛(𝑥) have the generating function^∞_𝑛=0𝐵_𝑛(𝑥)𝑡^𝑛𝑛!=𝑒^{𝑥(𝑒𝑡−1)}. (4.2) They also satisfy the following remarkable Dobinski-type formula𝐵_𝑛(𝑥)=𝑒^∞−𝑥_𝑖=0𝑖^𝑛𝑥𝑖!^𝑖, (4.3) which reduces to the Dobinski formula when 𝑥=1. It is worth noting that 𝐵_𝑛(𝑥) is represented as an infinite series in 𝑖.

As we know, the Dobinski-type formulas have been the subject of much combinatorial interest. Thus, it is worth looking for a general Dobinski-type formula.

In this section, we define a generalized Bell polynomials by𝐵_{𝑛;𝜶,𝜽′,𝑟}(𝑥)=^𝑛_𝑘=0𝑆𝑛,𝑘;𝜶,𝜽^′,𝑟𝑥^𝑘, (4.4) where 𝜽^′=(0,𝜃^′,…) and 𝜃^′≠0. Naturally, one get an extended definition of generalized Bell numbers as follow:𝐵_{𝑛;𝜶,𝜽′,𝑟}=^𝑛_𝑘=0𝑆𝑛,𝑘;𝜶,𝜽^′,𝑟. (4.5) Note that 𝐵_{𝑛;𝟎,𝟏,0}(𝑥)=𝐵_𝑛(𝑥) and 𝐵_{𝑛;𝟎,𝟏,0}=𝐵_𝑛. We can make use of (3.6) to obtain the following Dobinski-type formula.

Theorem 4.1. For 𝜽^′=(0,𝜃^′,…) and arbitrary 𝜶,𝑟, we have the Dobinski-type formula 𝐵_{𝑛;𝜶,𝜽′,𝑟}(𝑥)=𝑒^{−𝑥/𝜃′∞}_𝑖=0𝜔_𝑛,𝜶𝑟+𝑖𝜃^′𝑥𝑖!𝜃^′𝑖, (4.6) where 𝜔_𝑛,𝜶 is defined by (2.1).

Proof. By (4.4) we have ^∞_𝑛=0𝐵_{𝑛;𝜶,𝜽′,𝑟}(𝑥)𝑡^𝑛=𝑛!^∞_𝑛=0𝑡^𝑛𝑛!^𝑛_𝑘=0𝑆𝑛,𝑘;𝜶,𝜽^′,𝑟𝑥^𝑘=^∞_𝑘=0𝑥^𝑘_𝑛≥𝑘𝑆𝑛,𝑘;𝜶,𝜽^′𝑡,𝑟^𝑛.𝑛! (4.7) Replacing 𝜷 by 𝜽^′ in (3.6) yields ^∞_𝑛=0𝐵_{𝑛;𝜶,𝜽′,𝑟}(𝑥)𝑡^𝑛=𝑛!^∞_𝑘=0𝑥^𝑘_𝑛≥𝑘Δ𝑟,𝑟+𝜃^′,…,𝑟+𝑘𝜃^′𝜔_𝑛,𝜶𝑡^𝑛=𝑛!_∞𝑛=0𝑡^𝑛𝑛!^∞_𝑘=01𝑥𝑘!𝜃^′𝑘𝑘_𝑖=0(−1)^𝑘−𝑖𝑘𝑖𝜔_𝑛,𝜶𝑟+𝑖𝜃′. (4.8) By equating the coefficient of 𝑡^𝑛/𝑛! within the first and last expressions, we arrive at 𝐵_{𝑛;𝜶,𝜽′,𝑟}(𝑥)=^∞_𝑘=01𝑥𝑘!𝜃^′𝑘𝑘_𝑖=0(−1)^𝑘−𝑖𝑘𝑖𝜔_𝑛,𝜶𝑟+𝑖𝜃′. (4.9) Using the Cauchy product rule gives 𝐵_{𝑛;𝜶,𝜽′,𝑟}(𝑥)=^∞_𝑗=0(−1)^𝑗𝑥𝑗!𝜃^′𝑗∞_𝑖=0𝜔_𝑛,𝜶𝑟+𝑖𝜃^′𝑥𝑖!𝜃^′𝑖. (4.10) This implies (4.6) is true and completes the proof.

Letting 𝑥=1 we directly obtain the generalized Dobinski formula.

Corollary 4.2. For 𝜽^′=(0,𝜃^′,…) and arbitrary 𝜶,𝑟, we have 𝐵_{𝑛;𝜶,𝜽′,𝑟}=𝑒^{−1/𝜃′∞}_𝑖=0𝜔_𝑛,𝜶𝑟+𝑖𝜃^′1𝑖!𝜃^′𝑖, (4.11) where 𝜔_𝑛,𝜶 is defined by (2.1).

It is clear that (4.3) is a special case of (4.6) with 𝜃^′=1,𝜶=𝟎,𝑟=0.

Let 𝜙_𝜶,𝜃′(𝑡)=∑^∞_𝑖=0(𝑖𝜃^′−𝛼₀)⋯(𝑖𝜃′−𝛼_𝑛−1)(𝑡^𝑖/𝑖!). It is worth noting that the formula (4.6) can be used to obtain a closed sum formula for this type of infinite series. As mentioned in [1], such a type of series cannot be summed by using the hypergeometric series method. Let 𝑡=𝑥/𝜃^′,𝑟=0; then according to (4.6) we have𝜙_𝜶,𝜃′(𝑡)=𝐵_{𝑛;𝜶,𝜽′,0}𝜃^′𝑡𝑒^𝑡=𝑒^𝑡𝑛_𝑘=0𝑆𝑛,𝑘;𝜶,𝜽^′,0𝜃^′𝑘𝑡^𝑘. (4.12)

Example 4.3. Letting 𝜶=𝜽=(0,𝜃,…) we immediately obtain the Dobinski-type formula due to Hsu and Shiue [1] as follows: 𝐵_{𝑛;𝜽,𝜽′,𝑟}(𝑥)=𝑒^{−𝑥/𝜃′∞}_𝑖=0𝜃^𝑛𝑟+𝑖𝜃^′/𝜃^(𝑛)𝑥𝑖!𝜃^′𝑖. (4.13)

Example 4.4. Letting 𝜶=𝐝,𝜃^′=1,𝑛=𝑠₁+𝑠₂+⋯+𝑠_𝑚 we have the following Dobinski-type formula due to Blasiak [18]: 𝐵_𝑛;𝐝_,𝟏,0(𝑥)=𝑒^∞−𝑥_𝑖=𝑠1^𝑚_𝑗=1𝑑_𝑗−1+𝑖^(𝑠𝑗)𝑥^𝑖𝑖!. (4.14)

Acknowledgments

The author thanks the anonymous referees for their valuable suggestions and comments. This work was supported by the Zhejiang Province Natural Science Foundation (Grant nos. Y6110310 and Y6100021) and the Ningbo Natural Science Foundation.