A Determinant Expression for the Generalized Bessel Polynomials

Abstract

Using the exponential Riordan arrays, we show that a variation of the generalized Bessel polynomial sequence is of Sheffer type, and we obtain a determinant formula for the generalized Bessel polynomials. As a result, the Bessel polynomial is represented as determinant the entries of which involve Catalan numbers.

1. Introduction

The Bessel polynomials form a set of orthogonal polynomials on the unit circle in the complex plane. They are important in certain problems of mathematical physics; for example, they arise in the study of electrical networks and when the wave equation is considered in spherical coordinates. Many other important applications of such polynomials may be found in Grosswald [1, pages 131–149]. The Bessel polynomials [1, 2] are the polynomial solutions of the second-order differential equations 𝑧²𝑦_𝑛^′′(𝑧)+(2𝑧+2)𝑦^′_𝑛(𝑧)=𝑛(𝑛+1)𝑦_𝑛(𝑧),𝑦_𝑛(0)=1. (1) From this differential equation one can easily derive the formula 𝑦_𝑛(𝑧)=^𝑛_𝑘=0𝑛𝑘𝑘𝑛+𝑘𝑧𝑘!^𝑘2^𝑘. (2) Let 𝑎(𝑛,𝑘) denote the coefficient of 𝑧^𝑛−𝑘 in 𝑦_𝑛−1(𝑧). We call 𝑎(𝑛,𝑘) a signless Bessel number of the first kind and 𝑏(𝑛,𝑘)=(−1)^𝑛−𝑘𝑎(𝑛,𝑘) a Bessel number of the first kind. The Bessel number of the second kind 𝐵(𝑛,𝑘) is defined to be the number of set partitions of [𝑛]∶={1,2,3,…,𝑛} into 𝑘 blocks of size one or two. Han and Seo [3] showed that the two kinds of Bessel numbers are related by inverse formulas, and both Bessel numbers of the first kind and those of the second kind form log-concave sequences. In [4], the Bessel matrices obtained from the Bessel numbers of the first kind 𝑏(𝑛,𝑘) and of the second kind 𝐵(𝑛,𝑘) are introduced, respectively. It is shown that these matrices can be represented by the exponential Riordan matrices. The relations between the Bessel matrices, Catalan matrices, Stirling matrices, and Fibonacci matrices are also investigated. Recently, [5] showed that Bessel numbers correspond to a special case of the generalized Stirling numbers. Bessel polynomials have the following two determinant representations [6]:2^𝑛𝑦_𝑛(𝑧)=2()()2()1+𝑛𝑧−10⋯0−2𝑧2+𝑛𝑧1+𝑛𝑧−2⋯02𝑧²()()2()2()3+𝑛𝑧−2𝑧2+𝑛𝑧1+𝑛𝑧⋯0⋮⋮⋮⋱⋮−𝑧^𝑛−1(𝑛+𝑛𝑧)2(−𝑧)^𝑛−2(𝑛−1+𝑛𝑧)2(−𝑧)^𝑛−3()(),𝑦𝑛−2+𝑛𝑧⋯21+𝑛𝑧_𝑛(𝑧)=()1()().2𝑛−1𝑧−10⋯0002𝑛−3𝑧−1⋯000012𝑛−5𝑧⋯000⋮⋮⋮⋱⋮⋮⋮000⋯1𝑧−1000⋯011 (3)In Egecioglu [7], a (𝑛+1)×(𝑛+1) Hankel determinant 𝐻_𝑛(𝑧) is introduced by 𝐻_𝑛(𝑧)=det[𝑒_𝑖+𝑗(𝑧)]_{0≤𝑖,𝑗≤𝑛}, where 𝑒_𝑘(𝑧)=∑^𝑘_𝑚=0(𝑧^𝑚/𝑚!), and a closed form evaluation of this determinant in terms of the Bessel polynomials is given. Hence another determinant expression for the Bessel polynomials is given by 𝑐_𝑛𝑦_𝑛(2𝑧)=𝐻_𝑛(𝑧) with 𝐻_𝑛(𝑧)=det[𝑎_𝑖+𝑗(𝑧)]_{0≤𝑖,𝑗≤𝑛}, where 𝑐_𝑛=(−1)^{𝑛(𝑛+1)/2}2^𝑛∏^𝑛_𝑗=1(𝑗!²/(2𝑗)!²) and 𝑎_𝑘(𝑧)=∑^𝑘_𝑚=0(𝑧^𝑚/(𝑘−𝑚)!).

In this paper, using the production matrix of Bessel matrix which is represented as exponential Riordan array, we show that a variation of the generalized Bessel polynomial sequence is of Sheffer type, and we give a determinant formula for the generalized Bessel polynomials. As a result, the Bessel polynomial is represented as determinant the entries of which involve Catalan numbers.

2. Exponential Riordan Array

The concept of Riordan array was introduced by Shapiro et al. [8] in 1991; then the concept is generalized to the exponential Riordan array by many authors [4, 9–13]. Let 𝑔(𝑡)=1+∑^∞_𝑛=1𝑔_𝑛(𝑡^𝑛/𝑛!), ℎ(𝑡)=∑^∞_𝑛=1ℎ_𝑛(𝑡^𝑛/𝑛!) with ℎ₁≠0 be two formal power series. An exponential Riordan array is an infinite lower triangular array 𝐴=(𝑎_𝑛,𝑘)_{𝑛,𝑘≥0} whose 𝑘th column has exponential generating function (1/𝑘!)𝑔(𝑡)ℎ(𝑡)^𝑘. The array corresponding to the pair 𝑔(𝑡), ℎ(𝑡) is denoted by 𝐴=[𝑔(𝑡),ℎ(𝑡)]. The group law is then given by [𝑔(𝑡),ℎ(𝑡)][𝑑(𝑡),𝑙(𝑡)]=[𝑔(𝑡)𝑑(ℎ(𝑡)),𝑙(ℎ(𝑡))]. From the definition it follows at once that the bivariate generating function of 𝐴=[𝑔(𝑡),ℎ(𝑡)], namely, 𝐺_𝐴(𝑥,𝑡)=∑_𝑛,𝑘𝑎_𝑛,𝑘𝑥^𝑘(𝑡^𝑛/𝑛!), is given by 𝐺_𝐴(𝑥,𝑡)=𝑔(𝑡)exp𝑥𝑓(𝑡). (4)

Each exponential Riordan array 𝐴=[𝑔(𝑡),ℎ(𝑡)] is associated a matrix 𝑃 called its production matrix [12], which is determined by 𝑃=𝐴⁻¹𝐴, where 𝐴 is the matrix 𝐴 with its top row removed. The production matrix of an exponential Riordan array is characterized by the following lemma [10].

Lemma 1. Let 𝐴=(𝑎_𝑛,𝑘)_{𝑛,𝑘≥0}=[𝑔(𝑡),ℎ(𝑡)] be an exponential Riordan matrix and let 𝑐(𝑡)=∑^∞_𝑛=0𝑐_𝑛𝑡^𝑛, 𝑟(𝑡)=∑^∞_𝑛=0𝑟_𝑛𝑡^𝑛 be two formal power series such that 𝑟(ℎ(𝑡))=ℎ^′(𝑡), and 𝑐(ℎ(𝑡))=𝑔^′(𝑡)/𝑔(𝑡). Then 𝑎_𝑛+1,𝑘=(1/𝑘!)∑_{𝑖≥𝑘−1}𝑖!(𝑐_𝑖−𝑘+𝑘𝑟_{𝑖−𝑘+1})𝑎_𝑛,𝑖, where defining 𝑐₋₁=0. Conversely, starting from the sequences 𝑐(𝑡)=∑^∞_𝑛=0𝑐_𝑛𝑡^𝑛, and 𝑟(𝑡)=∑^∞_𝑛=0𝑟_𝑛𝑡^𝑛, the infinite array (𝑎_𝑛,𝑘)_{𝑛,𝑘≥0} defined by above relations is an exponential Riordan matrix.

From this proposition, the production matrix of 𝐴=[𝑔(𝑡),ℎ(𝑡)] is 𝑃=(𝑝_𝑖,𝑗)_{𝑖,𝑗≥0} with 𝑝_𝑖,𝑗=(𝑖!/𝑗!)(𝑐_𝑖−𝑗+𝑗𝑟_{𝑖−𝑗+1}) (𝑐₋₁=0), where the numbers 𝑐_𝑛 and 𝑟_𝑛 are determined by 𝑐(𝑡)=∑^∞_𝑛=0𝑐_𝑛𝑡^𝑛=𝑔^′(𝑓(𝑡))/𝑔(𝑓(𝑡)), 𝑟(𝑡)=∑^∞_𝑛=0𝑟_𝑛𝑡^𝑛=𝑓^′(𝑓(𝑡)). The sequence (𝑐_𝑛)_𝑛≥0 and (𝑟_𝑛)_𝑛≥0 are called, respectively, the 𝑐-sequence and the 𝑟-sequence of exponential Riordan array 𝐴=[𝑔(𝑡),ℎ(𝑡)].

As well known, many polynomial sequences like Laguerre polynomials, first and second kind Meixner polynomials, Poisson-Charlier polynomials, and Stirling polynomials form Sheffer sequences. The Sheffer polynomials are specified here by means of the following generating function (see [14, 15] and the references cited therein): ^∞_𝑛=0𝑠_𝑛(𝑥)𝑡^𝑛=1𝑛!𝑔𝑓(𝑡)𝑒^{𝑥𝑓(𝑡)}, (5) where 𝑓(𝑡) is a delta series and 𝑔(𝑡) an invertible series, and 𝑓(𝑡) is the compositional inverse of 𝑓(𝑡). The sequence 𝑠_𝑛(𝑥) in (5) is called the Sheffer sequence for the pair (𝑔(𝑡),𝑓(𝑡)). The Sheffer sequence, 𝑝_𝑛(𝑥) for (1,𝑓(𝑡)) is called the associated sequence and it satisfies binomial formula 𝑝_𝑛(𝑥+𝑦)=∑^𝑛_𝑘=0𝑝_𝑘(𝑥)𝑝_𝑛−𝑘(𝑦), so we also say that it is of binomial type. In [13], the author proved the following theorem ([13, Theorem 2.4]).

Lemma 2. Let [𝑔(𝑡),𝑓(𝑡)] be an exponential Riordan array with the 𝑐-sequences (𝑐_𝑖)_𝑖≥0 and 𝑟-sequences (𝑟_𝑖)_𝑖≥0. Let (𝑎_𝑛(𝑥))_𝑛≥0 be the Sheffer polynomial sequence for (𝑔(𝑡),𝑓(𝑡)). Then 𝑎₀(𝑥)=1, 𝑎₁(𝑥)=𝑥−𝑐₀, and for all 𝑛≥0, 𝑎_𝑛+1(𝑥)=det(𝑥𝐼_𝑛+1−𝑃_𝑛+1), where 𝑃_𝑛+1 is the (𝑛+1)th principal submatrix of the production matrix 𝑃.

3. Bessel Polynomials and Bessel Matrices

For 𝑛≥𝑘≥0, the signless Bessel number of the first kind 𝑎(𝑛,𝑘) is defined to be the coefficient of 𝑧^𝑛−𝑘 in Bessel polynomial 𝑦_𝑛−1(𝑧) with 𝑎(0,0)=1. The infinite lower triangular matrix 𝐴 with generic entry 𝑎(𝑛,𝑘) is called signless Bessel matrix of the first kind. From [4], the signless Bessel matrix of the first kind can be represented as the exponential Riordan array 𝐴=[1,1−√1−2𝑡].

Let 𝑝_𝑛(𝑥)=𝑥^𝑛𝑦_𝑛−1(1/𝑥) for 𝑛≥1 and 𝑝₀(𝑥)=1. Then the coefficients matrix of polynomial sequence {𝑝_𝑛(𝑥)}_𝑛≥0 is the signless Bessel matrix of the first kind 𝐴=[1,1−√1−2𝑡]. Let 𝜃_𝑛(𝑥)=𝑥^𝑛𝑦_𝑛(1/𝑥) for 𝑛≥0. Then 𝜃_𝑛(𝑥)=(1/𝑥)𝑝_𝑛+1(𝑥), and 𝜃_𝑛(𝑥)=𝑥^𝑛𝑦_𝑛(1/𝑥)=∑^𝑛_𝑘=0𝑎(𝑛+1,𝑘+1)𝑥^𝑘. By [10], the infinite lower triangular matrix 𝐵 with generic entry 𝐵_𝑛,𝑘=𝑎(𝑛+1,𝑘+1) is the exponential Riordan array 𝐵=[1/√1−2𝑡,1−√1−2𝑡]. We call this matrix the Bessel matrix since it is the coefficients matrix of the Bessel polynomials. From (4), one has ^∞_𝑛=0𝑝_𝑛(𝑥)𝑡^𝑛𝑛!=𝑒^{√𝑥(1−1−2𝑡)},_∞𝑛=0𝜃_𝑛(𝑥)𝑡^𝑛=1𝑛!√𝑒1−2𝑡^{√𝑥(1−1−2𝑡)}. (6)(7) Hence we obtain the following results (see also [14, 16]).

Theorem 3. The polynomial sequence {𝑝_𝑛(𝑥)}_𝑛≥0 is of binomial type, and the polynomial sequence {𝜃_𝑛(𝑥)}_𝑛≥0 is a Sheffer sequence.

The generalized Bessel polynomial 𝑦_𝑛(𝑧,𝑎,𝑏) was defined by Krall and Frink [2] as the polynomial solution of the differential equation 𝑧²𝑦_𝑛^′′(𝑧)+(𝑎𝑧+𝑏)𝑦^′_𝑛(𝑧)=𝑛(𝑛+𝑎−1)𝑦_𝑛(𝑧),𝑏≠0,𝑎≠0,−1,−2,…. (8) The explicit expression of the generalized Bessel polynomials is 𝑦_𝑛(𝑧,𝑎,𝑏)=^𝑛_𝑘=0𝑛𝑘𝑘𝑛+𝑘+𝑎−2𝑧𝑘!𝑏^𝑘. (9) In addition these polynomials satisfy the three-term recursion relation(𝑛+𝑎−1)(2𝑛+𝑎−2)𝑦_𝑛+1(𝑧)=[(2𝑛+𝑎)(2𝑛+𝑎−2)(𝑧/𝑏)+𝑎−2](2𝑛+𝑎−1)𝑦_𝑛(𝑧)+𝑛(2𝑛+𝑎)𝑦_𝑛−1(𝑧), 𝑛≥2, with initial conditions 𝑦₀(𝑧)=1 and 𝑦₁(𝑧)=1+𝑎(𝑧/𝑏).

We introduce a two-variable polynomial sequence 𝑝_𝑛(𝑧,𝑥) by 𝑝_𝑛(𝑧,𝑥)=(2𝑧)^𝑛𝑝_𝑛(𝑥/2𝑧). Then 𝑝_𝑛(𝑧,𝑥)=∑^𝑛_𝑘=0𝑎(𝑛,𝑘)2^𝑛−𝑘𝑧^𝑛−𝑘𝑥^𝑘=∑^𝑛_𝑘=0^𝑛−1_𝑘−1^{2𝑛−𝑘−1}_𝑛−1(𝑛−𝑘)!𝑧^𝑛−𝑘𝑥^𝑘.

Theorem 4. {𝑝_𝑛(𝑧,𝑥)}_𝑛≥0 is a sequence of polynomials of binomial type in the variable 𝑥.

Proof. From Theorem 3, we obtain ^∞_𝑛=0𝑝_𝑛()𝑡𝑧,𝑥^𝑛=𝑛!^∞_𝑛=0(2𝑧)^𝑛𝑝_𝑛𝑥𝑡2𝑧^𝑛=𝑛!_∞𝑛=0𝑝_𝑛𝑥()2𝑧2𝑧𝑡^𝑛𝑛!=𝑒^{√𝑥((1−1−4𝑧𝑡)/2𝑧)}. (10) Using (5), we obtain the desired result.

From the previous proof, the polynomial sequence 𝑝_𝑛(𝑧,𝑥) has exponential generating function ^∞_𝑛=0𝑝_𝑛()𝑡𝑧,𝑥^𝑛𝑛!=𝑒^{√𝑥((1−1−4𝑧𝑡)/2𝑧)}, (11) and 𝑝_𝑛(𝑥)=𝑝_𝑛(1/2,𝑥). So {𝑝_𝑛(𝑧,𝑥)}_𝑛≥0 is a generalization of {𝑝_𝑛(𝑥)}_𝑛≥0.

Instead of working with 𝑦_𝑛(𝑧,𝑎,𝑏), it is somewhat easier to work with the polynomials 𝑠_𝑛(𝑧,𝑎,𝑥) defined as 𝑠_𝑛(𝑧,𝑎,𝑥)=∑^𝑛_𝑘=0(^𝑛_𝑘)_𝑘^{𝑛+𝑘+𝑎−2}𝑘!𝑧^𝑘𝑥^𝑛−𝑘, which are related to the 𝑦_𝑛(𝑧,𝑎,𝑏) by 𝑠_𝑛(𝑧,𝑎,𝑥)=𝑥^𝑛𝑦_𝑛(𝑧,𝑎,𝑥), 𝑛≥0. It is easy to check that 𝑠_𝑛(1/2,2,𝑥)=𝜃_𝑛(𝑥). So {𝑠_𝑛(𝑧,𝑎,𝑥)}_𝑛≥0 is a generalization of {𝜃_𝑛(𝑥)}_𝑛≥0.

Theorem 5. {𝑠_𝑛(𝑧,𝑎,𝑥)}_𝑛≥0 is a Sheffer polynomial sequence in the variable 𝑥.

Proof. Let 𝐵(𝑧)=[(1/√1−4𝑧𝑡)((1−√1−4𝑧𝑡)/2𝑧𝑡)^𝑎−2,(1−√1−4𝑧𝑡)/2𝑧]. By (5.72) of [17, page 203], one has the general term of 𝐵(𝑧) which is 𝐵(𝑧)_𝑛,𝑘??=??[𝑡^𝑛/𝑛!](1/√1−4𝑧𝑡)𝑡^𝑘((1−√1−4𝑧𝑡)/2𝑧𝑡)^{𝑘+𝑎−2}??=??[𝑡^𝑛/𝑛!]𝑡^𝑘∑^∞_𝑛=0𝑛^{2𝑛+𝑘+𝑎−2}(𝑧𝑡)^𝑛??=??(^𝑛_𝑘)^{2𝑛−𝑘+𝑎−2}_𝑛−𝑘(𝑛−𝑘)!𝑧^𝑛−𝑘. Hence ∑^𝑛_𝑘=0𝐵(𝑧)_𝑛,𝑘𝑥^𝑘??=??∑^𝑛_𝑘=0(^𝑛_𝑘)^{2𝑛−𝑘+𝑎−2}_𝑛−𝑘(𝑛−𝑘)!𝑧^𝑛−𝑘𝑥^𝑘??=??∑^𝑛_𝑘=0(^𝑛_𝑛−𝑘)^{2𝑛−𝑘+𝑎−2}_𝑛−𝑘(𝑛−𝑘)!𝑧^𝑛−𝑘𝑥^𝑘??=??∑^𝑛_𝑘=0(^𝑛_𝑘)_𝑘^{𝑛+𝑘+𝑎−2}𝑘!𝑧^𝑘𝑥^𝑛−𝑘??=??𝑥^𝑛∑^𝑛_𝑘=0(^𝑛_𝑘)_𝑘^{𝑛+𝑘+𝑎−2}𝑘!(𝑧/𝑥)^𝑘??=??𝑥^𝑛𝑦_𝑛(𝑧,𝑎,𝑥)??=??𝑠_𝑛(𝑧,𝑎,𝑥). Therefore, the exponential generating function of the sequence {𝑠_𝑛(𝑧,𝑎,𝑥)}_𝑛≥0 is ^∞_𝑛=0𝑠_𝑛()𝑡𝑧,𝑎,𝑥^𝑛=1𝑛!√√1−4𝑧𝑡1−1−4𝑧𝑡2𝑧𝑡^𝑎−2×𝑒^{√𝑥((1−1−4𝑧𝑡)/2𝑧)}. (12) Using (5), we obtain the desired result.

From Theorems 4 and 5, we get the following theorem.

Theorem 6. The sequences {𝑝_𝑛(𝑧,𝑥)}_𝑛≥0 and {𝑠_𝑛(𝑧,𝑎,𝑥)}_𝑛≥0 satisfy the following convolution formulas: 𝑝_𝑛𝑧,𝑥+𝑦=^𝑛_𝑘=0𝑝_𝑘(𝑧,𝑥)𝑝_𝑛−𝑘,𝑠𝑧,𝑦_𝑛𝑧,𝑎,𝑥+𝑦=^𝑛_𝑘=0𝑠_𝑘(𝑧,𝑎,𝑥)𝑝_𝑛−𝑘𝑧,𝑦. (13)

Note that the coefficients matrix of the polynomial sequence {𝑝_𝑛(𝑧,𝑥)}_𝑛≥0 is 𝐴(𝑧)=[1,(1−√1−4𝑧𝑡)/2𝑧], which is a generalization of the signless Bessel matrix of the first kind 𝐴=[1,1−√1−2𝑡] while the coefficients matrix of the polynomial sequence {𝑠_𝑛(𝑧,𝑎,𝑥)}_𝑛≥0 is 𝐵(𝑧,𝑎)=[(1/√1−4𝑧𝑡)((1−√1−4𝑧𝑡)/2𝑧𝑡)^𝑎−2,(1−√1−4𝑧𝑡)/2𝑧], which is a generalization of the Bessel matrix 𝐵=[1/√1−2𝑡,1−√1−2𝑡].

4. Determinant Formulae for Bessel Polynomials

The Catalan numbers, 𝐶_𝑛=(1/(𝑛+1))_𝑛^2𝑛, may be defined by the ordinary generating function ∑^∞_𝑛=0𝐶_𝑛𝑡^𝑛=(1−√1−4𝑡)/2𝑡. The formulae (11) and (12) imply the connection between the generalized Bessel polynomials and Catalan numbers. In the following we will give some determinant expressions for Bessel polynomials, in which the entries involve Catalan numbers.

Define 𝑃(𝑧,𝑎) to be the infinite Hessenberg matrix with general entry 𝑝_𝑖,𝑗(𝑧,𝑎)=−1,if𝑖=𝑗−1,𝑖!()𝐶𝑗!2𝑖+𝑎_𝑖−𝑗𝑧^{𝑖−𝑗+1},if𝑖≥𝑗≥0,0,otherwise. (14)

The first rows of 𝑃 are𝑃(𝑧,𝑎)=−𝑎𝐶₀−()𝐶𝑧100⋯2+𝑎₁𝑧²−(2+𝑎)𝐶₀()𝐶𝑧10⋯−24+𝑎₂𝑧³−2(4+𝑎)𝐶₁𝑧²−(4+𝑎)𝐶₀()𝐶𝑧1⋯−66+𝑎₃𝑧⁴−6(6+𝑎)𝐶₂𝑧³−3(6+𝑎)𝐶₁𝑧²−(6+𝑎)𝐶₀()𝐶𝑧⋯−248+𝑎₄𝑧⁵−24(8+𝑎)𝐶₃𝑧⁴−12(8+𝑎)𝐶₂𝑧³−4(8+𝑎)𝐶₁𝑧²⋯⋮⋮⋮⋮⋱. (15)

Theorem 7. For 𝑛=0,1,2,…, the polynomial 𝑠_𝑛+1(𝑧,𝑎,𝑥) admits the following determinant representation 𝑠_𝑛+1(𝑧,𝑎,𝑥)=det(𝑥𝐼_𝑛+1−𝑃_𝑛+1(𝑧,𝑎)), that is,𝑠_𝑛+1(𝑧,𝑎,𝑥)=𝑥+𝑎𝐶₀()𝐶𝑧−10⋯02+𝑎₁𝑧²𝑥+(2+𝑎)𝐶₀2()𝐶𝑧−1⋯04+𝑎₂𝑧³2(4+𝑎)𝐶₁𝑧²𝑥+(4+𝑎)𝐶₀()𝑧⋯0⋮⋮⋮⋱⋮𝑛!2𝑛+𝑎𝐶0!^𝑛𝑧^𝑛+1𝑛!(2𝑛+𝑎)𝐶1!^𝑛−1𝑧^𝑛𝑛!(2𝑛+𝑎)𝐶2!^𝑛−2𝑧^𝑛−1⋯𝑥+(2𝑛+𝑎)𝐶₀𝑧, (16)where 𝑃_𝑛+1(𝑧,𝑎) is the (𝑛+1)th principal submatrix of Hessenberg matrix 𝑃(𝑧,𝑎) defined by (14).

Proof. Let 𝑔(𝑡)=(1−2𝑧𝑡)(1−𝑧𝑡)^𝑎−2 and 𝑓(𝑡)=𝑡(1−𝑧𝑡); then 𝑓(𝑡)=(1−√1−4𝑧𝑡)/2𝑧, 1/𝑔(𝑓(𝑡))=(1/√1−4𝑧𝑡)((1−√1−4𝑧𝑡)/2𝑧𝑡)^𝑎−2. Therefore, from (12), {𝑠_𝑛(𝑧,𝑎,𝑥)}_𝑛≥0 is a Sheffer polynomial sequence for the pair ((1−2𝑧𝑡)(1−𝑧𝑡)^𝑎−2,𝑡(1−𝑧𝑡)).
The generating functions for the 𝑟-sequence and 𝑐-sequence of [𝑔(𝑡),𝑓(𝑡)] are 𝑟(𝑡)??=??𝑓^′(𝑓(𝑡))??=??√1−4𝑧𝑡??=??1−2∑^∞_𝑛=1𝐶_𝑛−1𝑧^𝑛𝑡^𝑛, 𝑐(𝑡)??=??𝑔^′(𝑓(𝑡))/𝑔(𝑓(𝑡))??=??(2−𝑎)𝑧𝐶(𝑧𝑡)−2𝑧/(√1−4𝑧𝑡)??=??∑^∞_𝑛=0−(2𝑛+𝑎)𝐶_𝑛𝑧^𝑛+1𝑡^𝑛. By Lemma 1, the production matrix of [𝑔(𝑡),𝑓(𝑡)] is exactly the Hessenberg matrix 𝑃 defined by (14). Hence, by Lemma 2, we obtain the desired result.

Corollary 8. For 𝑛=0,1,2,…, one has 𝑦_𝑛+1(𝑧,𝑎,𝑥)=det(𝐼_𝑛+1−(1/𝑥)𝑃_𝑛+1(𝑧,𝑎)). Particularly,𝑦_𝑛+1(𝑧)=12^𝑛+12+2𝐶₀𝑧−10⋯02!0!⋅2𝐶₁𝑧²2+4𝐶₀𝑧−1⋯03!0!⋅2𝐶₂𝑧³3!1!⋅2𝐶₁𝑧²2+6𝐶₀()!𝑧⋯0⋮⋮⋮⋱⋮𝑛+10!⋅2𝐶_𝑛𝑧^𝑛+1(𝑛+1)!1!⋅2𝐶_𝑛−1𝑧^𝑛(𝑛+1)!2!⋅2𝐶_𝑛−2𝑧^𝑛−1⋯2+2(𝑛+1)𝐶₀𝑧. (17)

Corollary 9. For 𝑛=0,1,2,…, we have𝜃_𝑛+1(𝑥)=𝑥+𝐶₀−10⋯02!𝐶0!2¹𝑥+2𝐶₀−1⋯03!0!2²𝐶₂3!𝐶1!2¹𝑥+3𝐶₀()!⋯0⋮⋮⋮⋱⋮𝑛+10!2^𝑛𝐶_𝑛(𝑛+1)!1!2^𝑛−1𝐶_𝑛−1(𝑛+1)!2!2^𝑛−2𝐶_𝑛−2⋯𝑥+(𝑛+1)𝐶₀,𝑦_𝑛+1(𝑥)=1+𝐶₀𝑥−𝑥0⋯02!𝐶0!2¹𝑥1+2𝐶₀𝑥−𝑥⋯03!0!2²𝐶₂𝑥3!𝐶1!2¹𝑥1+3𝐶₀()!𝑥⋯0⋮⋮⋮⋱⋮𝑛+10!2^𝑛𝐶_𝑛𝑥(𝑛+1)!1!2^𝑛−1𝐶_𝑛−1𝑥(𝑛+1)!2!2^𝑛−2𝐶_𝑛−2𝑥⋯1+(𝑛+1)𝐶₀𝑥. (18)

Examples. (1) One has 𝑠₃()=𝑧,𝑎,𝑥𝑥+𝑎𝐶₀()𝐶𝑧−102+𝑎₁𝑧²𝑥+(2+𝑎)𝐶₀2()𝐶𝑧−14+𝑎₂𝑧³2(4+𝑎)𝐶₁𝑧²𝑥+(4+𝑎)𝐶₀𝑧=𝑥³+3(𝑎+2)𝑧𝑥²+3(𝑎+2)(𝑎+3)𝑧²𝑥+(𝑎+2)(𝑎+3)(𝑎+4)𝑧³. (19)

(2) One has 𝑦₃()=1𝑧,𝑎,𝑥𝑥³𝑥+𝑎𝐶₀()𝐶𝑧−102+𝑎₁𝑧²𝑥+(2+𝑎)𝐶₀2()𝐶𝑧−14+𝑎₂𝑧³2(4+𝑎)𝐶₁𝑧²𝑥+(4+𝑎)𝐶₀𝑧(𝑧=1+3𝑎+2)𝑥(𝑧+3𝑎+2)(𝑎+3)𝑥²+()𝑧𝑎+2)(𝑎+3)(𝑎+4𝑥³. (20)

(3) Consider the following: 𝑦₃(𝑧)=12³2+2𝑧−104𝑧²2+4𝑧−124𝑧³12𝑧²2+6𝑧=1+6𝑧+15𝑧²+15𝑧³. (21)

(4) Consider the following: 𝜃₃(𝑥)=𝑥+1−101𝑥+2−133𝑥+3=𝑥³+6𝑥²+15𝑥+15. (22)

(5) One has 𝑦₃(𝑥)=1+𝑥−𝑥0𝑥1+2𝑥−𝑥3𝑥3𝑥1+3𝑥=15𝑥³+15𝑥²+6𝑥+1. (23)

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant no. 11261032) and the Natural Science Foundation of Gansu Province (Grant no. 1010RJZA049).