Abstract

We employ the moment representations for Al-Salam-Carlitz polynomials and show how to deduce bilinear, trilinear, and multilinear generating functions for Al-Salam-Carlitz polynomials. Moreover, we obtain two terminating generating functions for Al-Salam-Carlitz polynomials by the method of moments.

Dedicated to Professor H. M. Srivastava

1. Introduction

The Al-Salam-Carlitz polynomials [1, equations and ] are important -orthogonal polynomials whose applications varies in many fields such as the -harmonic oscillator, theta function, quantum groups, and coding theory [24]. The generalized Al-Salam-Carlitz polynomials [5, equation ] and are defined by the following where , and There are close relationships between Al-Salam-Carlitz polynomials and other polynomials, such as -Bessel polynomials [6], Stieltjes-Wigert polynomials [7], and Rogers-Szegö polynomials [5]. For more information, please refer to [57].

Al-Salam and Carlitz [1] defined moments of two discrete distribution and by Rogers-Szegö polynomials as follows where is a step function whose jumps occur at the points and for , while the jumps of occur at the points for . These jumps are given by the following

In view of the fact that the discrete probability measure of moments in (1.4) on may be given by the sum of two terms [8, equation ], However, is given by [8, equation ], where and denotes a unit mass supported at , see also [9, 10].

Liu gained the following expression of bivariate Rogers-Szegö polynomials by the technique of partial fraction [11, equation ], So it's natural to define the generalized discrete probability measure and by the following where the bivariate Rogers-Szegö polynomials expressed by the following and their generating functions given by [11, equations and ],

Al-Salam and Ismail [9] built relations between multilinear generating functions for Rogers-Szegö polynomials and by the method of moments. Berg and Ismail [8] gained orthogonality relations of generating functions for Rogers-Szegö polynomials by moments. Ismail and Masson [12] obtained several -beta integrals and Bailey's from -Hermite polynomials by moments. Ismail and Stanton [13, 14] derived generating functions for Al-Salam-Chihara and continuous -Hermite polynomials by moments. For more information, please refer to [8, 9, 1218].

There are many ways to explore generating functions for Al-Salam-Carlitz polynomials. Carlitz [19] used combinatorial method. Srivastava and Jain [20] utilized the technique of transformation theory. Kim [21] obtained combinatorial interpretation by moments. Wang [22] employed the -integral representation. Chen et al. [5] applied -exponential operator. For more information, please refer to [2, 5, 1922].

One may ask naturally a question: can Al-Salam-Carlitz polynomials be expressed by moments and some related problems deduced by moments? In this paper, we would like to represent Al-Salam-Carlitz polynomials in terms of moments (see Theorem 2.3 below) and show how to deduce generating functions for Al-Salam-Carlitz polynomials by the method of moments.

This paper is organized as follows. In Section 3, two bilinear generating functions for Al-Salam-Carlitz polynomials are obtained by the method of moments. In Section 4, a trilinear generating function for Al-Salam-Carlitz polynomials is gained and a corollary is achieved. In Section 5, a multilinear generating function for Al-Salam-Carlitz polynomials is supplied naturally. In Section 6, two miscellaneous generating functions for Al-Salam-Carlitz polynomials are given.

2. Notations and Moments for Al-Salam-Carlitz Polynomials

In this paper, we follow the notations and terminology in [23] and suppose that . The -series and its compact factorials are defined, respectively, by the following and , where is a positive integer and is a nonnegative integer or . In the context, convergence of basic hypergeometric series is no issue at all because they are the terminating -series.

The -Chu-Vandermonde formula reads that [23, equations and ],

The -difference operators and defined by [23, 24],

Before the proof of main results, we need the following lemmas.

Lemma 2.1 (-Leibniz formula [25, 26]). For , one has

Lemma 2.2 (equations and ). For , one has for in (2.6) and in (2.7), one has

Theorem 2.3. For , one has where and are defined by (1.6) and (1.7).

Theorem 2.4. For and , if , one has
For , if , one has

Proof of Theorem 2.3. By (1.11) and (1.12), we have Differentiating by and using -Leibinz formula (2.5) for on both sides of (2.14), we have Comparing (1.2), (1.3), and (2.15), we have (2.11). Similarly, by -Leibniz formula (2.4) for , we obtain so we achieve (2.10) by (1.3). The proof is complete.

Proof of Theorem 2.4. Multiplying and summing over on both sides of (2.15) yields Differentiating by and using (2.5) on both sides of (2.17), we have that is Multiplying and summing on both sides of (2.19), we obtain (2.12) after simplification. Similarly, we have by -Leibniz formula for , we gain so we have which is (2.13) after multiplying and summing on both sides of (2.22). The proof is complete.

3. Bilinear Generating Function for Al-Salam-Carlitz Polynomials

Chen et al. [5] gave the following bilinear generating functions for Al-Salam-Carlitz polynomials.

Theorem 3.1 (see [5, Theorem 5.3]). Assume that . One has where and for nonnegative integers and .

In this section, we gain (3.1) and the following dual ones by the method of moments.

Theorem 3.2. Assume that . One has where and for nonnegative integers .

Corollary 3.3 (see [5, Theorem 4.5]). For , one has provided that or and .

Corollary 3.4. For , one has provided that or and .

Remark 3.5. For , formula (3.1) and (3.2) reduce to (3.3) and (3.4), respectively, by -Gauss sum [23, equation ], and -Chu-Vandermonde formula (2.2).

Proof of Theorems 3.1 and 3.2. The left-hand side of (3.1) is equal to which is the right hand side of (3.1) after simplification. Similarly, the right hand side of (3.2) is equivalent to which is the right hand side of (3.2) after using (2.9). The proof is complete.

4. Trilinear Generating Function for Al-Salam-Carlitz Polynomials

Verma and Jain deduced the following trilinear generating functions for Rogers-Szegö polynomials.

Proposition 4.1 (see [27, equation ]). For , one has

In this section, we obtain the following trilinear generating functions for Al-Salam-Carlitz polynomials.

Theorem 4.2. For , one has provided that the right hand side of (4.2) is convergent.

Remark 4.3. Ismail and Stanton [14] developed a method for deriving integral representations of certain orthogonal polynomials as moments and obtained [14, equation ] trilinear generating functions for -Hermite polynomials , which is equivalent to Proposition 4.1. For more information, please refer to [14].

Proof of Theorem 4.2. The left hand side of (4.2) is equal to which is the right hand of (4.2) after simplification. The proof is complete.

Proof of Proposition 4.1. Noting the fact that and If letting and in (4.2), the right hand side of (4.2) is equal to which reduces to the right hand side of (4.1) after setting . The proof is complete.

5. Multilinear Generating Functions for Al-Salam-Carlitz Polynomials

Ismail and Stanton gained the following multilinear generating functions for Rogers-Szegö polynomials.

Proposition 5.1 (see [28, equation ]). One has

In this section, we obtain the following multilinear generating functions for Al-Salam-Carlitz polynomials.

Theorem 5.2. For , one has provided that the right hand side of (5.2) is convergent.

Corollary 5.3 (see [20, equation ]). For , one has

Remark 5.4. Letting , and taking in Theorem 5.2, formula (5.2) reduces to (5.1). Setting and in Theorem 5.2, formula (5.2) reduces to (5.3).

Proof. The left hand side of (5.2) is equal to which is the right hand side of (5.2) after simplification. The proof is complete.

6. Some Miscellaneous Generating Functions for Al-Salam-Carlitz Polynomials

In this section, we deduce the following terminating generating functions for Al-Salam-Carlitz polynomials.

Theorem 6.1. For , one has

Theorem 6.2. For , one has

Corollary 6.3. For , one has

Remark 6.4. Letting in Theorems 6.1 and 6.2, noting the fact that Formula (6.1) and (6.2) reduce to -Chu-Vandermonde formula (2.2), respectively. Replacing by and letting in Theorems 6.1 and 6.2, formula (6.1) and (6.2) reduce to (6.3), respectively.

Proof of Theorems 6.1 and 6.2. Replacing by on both sides of the second formula in (2.2), we have Applying moments in (6.5), using (2.10) and (2.13), we gain