Moments for Generating Functions of Al-Salam-Carlitz Polynomials

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 (4.1) and (4.2)] ^∞_𝑛=0𝑈_𝑛^(𝑎)(𝑥)𝑡^𝑛𝑞;𝑞_𝑛=𝑡,𝑎𝑡;𝑞_∞𝑥𝑡;𝑞_∞,^∞_𝑛=0𝑉_𝑛^(𝑎)(𝑥)𝑡^𝑛𝑞;𝑞_𝑛=𝑥𝑡;𝑞_∞𝑡,𝑎𝑡;𝑞_∞ (1.1) are important 𝑞-orthogonal polynomials whose applications varies in many fields such as the 𝑞-harmonic oscillator, theta function, quantum groups, and coding theory [2–4]. The generalized Al-Salam-Carlitz polynomials 𝑈_𝑛(𝑥,𝑦,𝑎∣𝑞) [5, equation (1.2)] and 𝑉_𝑛(𝑥,𝑦,𝑎∣𝑞) are defined by the following ^∞_𝑛=0𝑈_𝑛𝑡𝑥,𝑦,𝑎∣𝑞^𝑛𝑞;𝑞_𝑛=𝑎𝑡,𝑦𝑡;𝑞_∞𝑥𝑡;𝑞_∞,^∞_𝑛=0𝑉_𝑛𝑡𝑥,𝑦,𝑎∣𝑞^𝑛𝑞;𝑞_𝑛=𝑥𝑡;𝑞_∞𝑎𝑡,𝑦𝑡;𝑞_∞, (1.2) where 𝑈_𝑛(𝑥,1,𝑎∣𝑞)=𝑈_𝑛^(𝑎)(𝑥), 𝑉_𝑛(𝑥,1,𝑎∣𝑞)=𝑉_𝑛^(𝑎)(𝑥) and 𝑈_𝑛𝑥,𝑦,𝑎∣𝑞=(−𝑎)^𝑛𝑞^𝑛2₂𝜙₁𝑞^−𝑛,𝑦𝑥0;𝑞,𝑞𝑥𝑎,𝑉_𝑛𝑥,𝑦,𝑎∣𝑞=𝑎^𝑛₂𝜙₀𝑞^−𝑛,𝑥𝑦;---𝑞,𝑦𝑞^𝑛𝑎. (1.3) There are close relationships between Al-Salam-Carlitz polynomials and other polynomials, such as 𝑞-Bessel polynomials [6], Stieltjes-Wigert polynomials [7], and Rogers-Szegö polynomials [5]. For more information, please refer to [5–7].

Al-Salam and Carlitz [1] defined moments of two discrete distribution d𝛼^(𝑎)(𝑥) and d𝛽^(𝑎)(𝑥) by Rogers-Szegö polynomials as follows ^∞_−∞𝑥^𝑛d𝛼^(𝑎)(𝑥)=ℎ_𝑛𝑎∣𝑞≜𝑉_𝑛0,1,𝑎∣𝑞_∞−∞,𝑥^𝑛d𝛽^(𝑎)(𝑥)=𝑔_𝑛𝑎∣𝑞≜(−1)^𝑛𝑞^𝑛2𝑈_𝑛0,1,𝑎∣𝑞, (1.4) where 𝛼^(𝑎)(𝑥) is a step function whose jumps occur at the points 𝑞^𝑘 and 𝑎𝑞^𝑘 for 𝑘∈ℕ, while the jumps of 𝛽^(𝑎)(𝑥) occur at the points 𝑞^−𝑘 for 𝑘∈ℕ. These jumps are given by the following d𝛼^(𝑎)𝑞^𝑘=𝑞^𝑘𝑎;𝑞_∞𝑞,𝑞/𝑎;𝑞_𝑘,d𝛼^(𝑎)𝑎𝑞^𝑘=𝑞^𝑘1/𝑎;𝑞_∞𝑞,𝑎𝑞;𝑞_𝑘,d𝛽^(𝑎)𝑞^−𝑘=𝑎^𝑘𝑞^𝑘2𝑎𝑞^𝑘+1;𝑞_∞𝑞;𝑞_𝑘. (1.5)

In view of the fact that the discrete probability measure 𝛼^(𝑎) of moments in (1.4) on [𝑎,1] may be given by the sum of two terms [8, equation (3.3)], 𝛼^(𝑎)=^∞_𝑘=0𝑞^𝑘𝑎;𝑞_∞𝑞,𝑞/𝑎;𝑞_𝑘𝜀_𝑞𝑘+𝑞^𝑘(1/𝑎;𝑞)_∞𝑞,𝑎𝑞;𝑞_𝑘𝜀_𝑎𝑞𝑘. (1.6) However, 𝛽^(𝑎) is given by [8, equation (3.15)], 𝛽^(𝑎)=𝑎𝑞;𝑞_∞^∞_𝑘=0𝑎^𝑘𝑞^𝑘2𝑎𝑞,𝑞;𝑞_𝑘𝜀_𝑞−𝑘, (1.7) where 𝑘∈ℕ and 𝜀_𝑦 denotes a unit mass supported at 𝑦, see also [9, 10].

Liu gained the following expression of bivariate Rogers-Szegö polynomials by the technique of partial fraction [11, equation (4.20)], ℎ_𝑛=𝑎𝑎,𝑏∣𝑞^𝑛𝑏/𝑎;𝑞_∞^∞_𝑘=0𝑞^(𝑛+1)𝑘𝑞,𝑎𝑞/𝑏;𝑞_𝑘+𝑏^𝑛𝑎/𝑏;𝑞_∞^∞_𝑘=0𝑞^(𝑛+1)𝑘𝑞,𝑞𝑏/𝑎;𝑞_𝑛. (1.8) So it's natural to define the generalized discrete probability measure 𝛼^(𝑎,𝑏) and 𝛽^(𝑎,𝑏) by the following 𝛼^(𝑎,𝑏)=^∞_𝑘=0𝑞^𝑘𝑎/𝑏;𝑞_∞𝑞,𝑞𝑏/𝑎;𝑞_𝑘𝜀_𝑏𝑞𝑘+𝑞^𝑘𝑏/𝑎;𝑞_∞𝑞,𝑎𝑞/𝑏;𝑞_𝑘𝜀_𝑎𝑞𝑘,𝛽^(𝑎,𝑏)=𝑎𝑞𝑏;𝑞_∞^∞_𝑘=0𝑎^𝑘𝑏^−𝑘𝑞^𝑘2𝑎𝑞/𝑏,𝑞;𝑞_𝑘𝜀_{𝑏𝑞−𝑘}, (1.9)(1.10) where the bivariate Rogers-Szegö polynomials expressed by the following ℎ_𝑛𝑎,𝑏∣𝑞=^+∞_−∞𝑥^𝑛d𝛼^(𝑎,𝑏)(𝑥)≜𝑉_𝑛,𝑔0,𝑎,𝑏∣𝑞_𝑛𝑎,𝑏∣𝑞=^+∞_−∞𝑥^𝑛d𝛽^(𝑎,𝑏)(𝑥)≜𝑈_𝑛0,𝑎,𝑏∣𝑞, (1.11) and their generating functions given by [11, equations (2.3) and (2.13)], ^∞_𝑛=0ℎ_𝑛𝑡𝑎,𝑏∣𝑞^𝑛𝑞;𝑞_𝑛=1𝑎𝑡,𝑏𝑡;𝑞_∞,^∞_𝑛=0𝑔_𝑛()𝑎,𝑏∣𝑞−1^𝑛𝑞^𝑛2𝑡^𝑛𝑞;𝑞_𝑛=𝑎𝑡,𝑏𝑡;𝑞_∞. (1.12)

Al-Salam and Ismail [9] built relations between multilinear generating functions for Rogers-Szegö polynomials and ₈𝑊₇ 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 𝑞-beta integrals and Bailey's ₆𝜓₆ from 𝑞-Hermite polynomials by moments. Ismail and Stanton [13, 14] derived generating functions for Al-Salam-Chihara and continuous 𝑞-Hermite polynomials by moments. For more information, please refer to [8, 9, 12–18].

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 𝑞-integral representation. Chen et al. [5] applied 𝑞-exponential operator. For more information, please refer to [2, 5, 19–22].

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 0<𝑞<1. The 𝑞-series and its compact factorials are defined, respectively, by the following 𝑎;𝑞₀=1,𝑎;𝑞_𝑛=^𝑛−1_𝑘=01−𝑎𝑞^𝑘,𝑎;𝑞_∞=^∞_𝑘=01−𝑎𝑞^𝑘 (2.1) and (𝑎₁,𝑎₂,…,𝑎_𝑚;𝑞)_𝑛=(𝑎₁;𝑞)_𝑛(𝑎₂;𝑞)_𝑛…(𝑎_𝑚;𝑞)_𝑛, where 𝑚 is a positive integer and 𝑛 is a nonnegative integer or ∞. In the context, convergence of basic hypergeometric series is no issue at all because they are the terminating 𝑞-series.

The 𝑞-Chu-Vandermonde formula reads that [23, equations (II.6) and (II.7)], ₂𝜙₁𝑞^−𝑛𝑐;=,𝑎𝑞,𝑞𝑐/𝑎;𝑞_𝑛𝑐;𝑞_𝑛𝑎^𝑛,₂𝜙₁𝑞^−𝑛𝑐;,𝑎𝑞,𝑐𝑞^𝑛𝑎=𝑐/𝑎;𝑞_𝑛𝑐;𝑞_𝑛. (2.2)

The 𝑞-difference operators 𝐷_𝑎 and 𝜃_𝑎 defined by [23, 24], 𝐷_𝑎𝑓(𝑎=𝑓(𝑎))−𝑓𝑎𝑞𝑎,𝜃_𝑎𝑓(𝑎=𝑓)𝑎𝑞⁻¹−𝑓(𝑎)𝑎𝑞⁻¹. (2.3)

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

Lemma 2.1 (𝑞-Leibniz formula [25, 26]). For 𝑛∈ℕ, one has 𝜃^𝑛_𝑎𝑓(𝑎)𝑔(𝑎)=^𝑛_𝑘=0𝑛𝑘𝜃^𝑘_𝑎𝑓(𝑎)𝜃_𝑎^𝑛−𝑘𝑔𝑎𝑞^−𝑘𝐷_𝑛𝑎,𝑓(𝑎)𝑔(𝑎)=^𝑛_𝑘=0𝑛𝑘𝑞^{𝑘(𝑘−𝑛)}𝐷^𝑘_𝑎𝑓(𝑎)𝐷_𝑎^𝑛−𝑘𝑔𝑎𝑞^𝑘. (2.4)(2.5)

Lemma 2.2 (equations (III.12) and (III.13)). For 𝑛∈ℕ, one has ₃𝜙₂𝑞^−𝑛;=,𝑏,𝑐𝑑,𝑒𝑞,𝑞𝑒/𝑐;𝑞_𝑛𝑐^𝑛𝑒;𝑞_𝑛³𝜙₂𝑞^−𝑛𝑑,𝑐,𝑏𝑑,𝑐𝑞^1−𝑛𝑒;𝑞,𝑏𝑞𝑒₃,𝜙₂𝑞^−𝑛;,𝑏,𝑐𝑑,𝑒𝑞,𝑑𝑒𝑞^𝑛=𝑏𝑐𝑒/𝑐;𝑞_𝑛𝑒;𝑞_𝑛³𝜙₂𝑞^−𝑛𝑑,𝑐,𝑏𝑑,𝑐𝑞^1−𝑛𝑒;𝑞,𝑞, (2.6)(2.7) for 𝑑→0 in (2.6) and 𝑑→∞ in (2.7), one has ₃𝜙₂𝑞^−𝑛;=,𝑏,𝑐0,𝑒𝑞,𝑞𝑒/𝑐;𝑞_𝑛𝑐^𝑛𝑒;𝑞_𝑛²𝜙₁𝑞^−𝑛,𝑐𝑐𝑞^1−𝑛𝑒;𝑞,𝑏𝑞𝑒₃,𝜙₁𝑞^−𝑛𝑒;,𝑏,𝑐𝑞,𝑒𝑞^𝑛=𝑏𝑐𝑒/𝑐;𝑞_𝑛𝑒;𝑞_𝑛²𝜙₁𝑞^−𝑛,𝑐𝑐𝑞^1−𝑛𝑒;𝑞𝑞,𝑏. (2.8)(2.9)

Theorem 2.3. For 𝑛∈ℕ, one has 𝑈_𝑛=()𝑥,𝑦,𝑎∣𝑞−1^𝑛𝑞^𝑛2𝑞𝑥/𝑦,𝑞𝑥/𝑎;𝑞_∞^∞_−∞𝜇^𝑛𝑞𝑥𝜇𝑎𝑦;𝑞_∞d𝛽^(𝑎,𝑦)𝜇,𝑉_𝑛=𝑥𝑥,𝑦,𝑎∣𝑞𝑦,𝑥𝑎;𝑞_∞^∞_−∞𝜇^𝑛𝑥𝜇/𝑎𝑦;𝑞_∞d𝛼^(𝑎,𝑦)𝜇, (2.10)(2.11) where 𝛼^(𝑦,𝑎)(𝜇) and 𝛽^(𝑦,𝑎)(𝜇) are defined by (1.6) and (1.7).

Theorem 2.4. For 𝑓∈ℕ and 𝑠/𝑟=𝑞^−𝑓, if max{|𝑎𝑟|,|𝑎𝑡|,|𝑏𝑡|,|𝑎𝑤|,|𝑏𝑤|}<1, one has ^∞_−∞𝑠𝜇;𝑞_∞𝑟𝜇,𝑡𝜇,𝑤𝜇;𝑞_∞d𝛼^(𝑎,𝑏)𝜇=𝑎𝑠,𝑎𝑏𝑡𝑤;𝑞_∞𝑎𝑟,𝑎𝑡,𝑏𝑡,𝑎𝑤,𝑏𝑤;𝑞_∞³𝜙₂𝑠𝑟;,𝑎𝑡,𝑎𝑤𝑎𝑠,𝑎𝑏𝑡𝑤𝑞,𝑏𝑟. (2.12)
For 𝑟/𝑠=𝑞^−𝑓, if max{|𝑎𝑠|,|𝑎𝑏𝑡𝑤/𝑞|}<1, one has ^∞_−∞𝑟𝜇,𝑡𝜇,𝑤𝜇;𝑞_∞𝑠𝜇;𝑞_∞d𝛽^(𝑎,𝑏)𝜇=𝑎𝑟,𝑎𝑡,𝑏𝑡,𝑎𝑤,𝑏𝑤;𝑞_∞𝑎𝑠,𝑎𝑏𝑡𝑤/𝑞;𝑞_∞³𝜙₂𝑟𝑠,𝑞(),𝑞𝑎𝑡()𝑞𝑎𝑤,𝑞𝑎𝑠²;𝑎𝑏𝑡𝑤𝑞,𝑞. (2.13)

Proof of Theorem 2.3. By (1.11) and (1.12), we have ^∞_−∞1𝑡𝜇;𝑞_∞d𝛼^(𝑎,𝑏)𝜇=1𝑎𝑡,𝑏𝑡;𝑞_∞. (2.14) Differentiating by 𝐷^𝑛_𝑡 and using 𝑞-Leibinz formula (2.5) for 𝐷_𝑡 on both sides of (2.14), we have ^∞_−∞𝜇^𝑛𝑡𝜇;𝑞_∞d𝛼^(𝑎,𝑏)𝜇=𝑎^𝑛𝑎𝑡,𝑏𝑡;𝑞_∞²𝜙₀𝑞^−𝑛;,𝑎𝑡---𝑞,𝑏𝑞^𝑛𝑎. (2.15) Comparing (1.2), (1.3), and (2.15), we have (2.11). Similarly, by 𝑞-Leibniz formula (2.4) for 𝜃_𝑡, we obtain ^∞_−∞𝜇^𝑛𝑡𝜇;𝑞_∞d𝛽^(𝑎,𝑏)𝜇=𝑎^𝑛𝑎𝑡,𝑏𝑡;𝑞_∞2𝜙₁𝑞^−𝑛,𝑞()0;𝑎𝑡𝑞,𝑏𝑡, (2.16) so we achieve (2.10) by (1.3). The proof is complete.

Proof of Theorem 2.4. Multiplying 𝑤^𝑛/(𝑞;𝑞)_𝑛 and summing 𝑛 over 0≤𝑛≤∞ on both sides of (2.15) yields ^∞_−∞1𝑡𝜇,𝑤𝜇;𝑞_∞d𝛼^(𝑎,𝑏)𝜇=𝑎𝑏𝑡𝑤;𝑞_∞𝑎𝑡,𝑏𝑡,𝑎𝑤,𝑏𝑤;𝑞_∞. (2.17) Differentiating by 𝐷^𝑛_𝑡 and using (2.5) on both sides of (2.17), we have 𝐷^𝑛_𝑡𝑎𝑏𝑡𝑤;𝑞_∞𝑎𝑡,𝑏𝑡;𝑞_∞=^𝑛_𝑘=0𝑞^{𝑘(𝑘−𝑛)}𝑛𝑘𝐷^𝑘_𝑡𝑎𝑏𝑡𝑤;𝑞_∞𝑏𝑡;𝑞_∞𝐷_𝑡^𝑛−𝑘1𝑎𝑡𝑞^𝑘;𝑞_∞=^𝑛_𝑘=0𝑞^{𝑘(𝑘−𝑛)}𝑛𝑘𝑏^𝑘𝑎𝑤;𝑞_𝑘𝑎𝑏𝑡𝑤𝑞^𝑘;𝑞_∞𝑏𝑡;𝑞_∞𝑎𝑞^{𝑘𝑛−𝑘}𝑎𝑡𝑞^𝑘;𝑞_∞, (2.18) that is ^∞_−∞𝜇^𝑛𝑡𝜇,𝑤𝜇;𝑞_∞d𝛼^(𝑎,𝑏)𝜇=𝑎^𝑛𝑎𝑏𝑡𝑤;𝑞_∞𝑎𝑡,𝑏𝑡,𝑎𝑤,𝑏𝑤;𝑞_∞³𝜙₁𝑞^−𝑛;,𝑎𝑤,𝑎𝑡𝑎𝑏𝑡𝑤𝑞,𝑏𝑞^𝑛𝑎. (2.19) Multiplying (𝑠/𝑟;𝑞)_𝑛𝑟^𝑛/(𝑞;𝑞)_𝑛 and summing on both sides of (2.19), we obtain (2.12) after simplification. Similarly, we have ^∞_−∞𝑡𝜇,𝑤𝜇;𝑞_∞d𝛽^(𝑎,𝑏)𝜇=𝑎𝑡,𝑏𝑡,𝑎𝑤,𝑏𝑤;𝑞_∞𝑎𝑏𝑡𝑤/𝑞;𝑞_∞, (2.20) by 𝑞-Leibniz formula for 𝜃_𝑡, we gain 𝜃^𝑛_𝑡𝑎𝑡,𝑏𝑡;𝑞_∞𝑎𝑏𝑡𝑤/𝑞;𝑞_∞=^𝑛_𝑘=0𝑛𝑘𝑞^−2𝑘+1(𝑎𝑏𝑤)^𝑘𝑞/(𝑎𝑤);𝑞_𝑘𝑏𝑡;𝑞_∞𝑎𝑏𝑡𝑤𝑞^−1−𝑘;𝑞_∞−𝑎𝑞^{−𝑘𝑛−𝑘}𝑎𝑡𝑞^−𝑘;𝑞_∞, (2.21) so we have ^∞_−∞𝜇^𝑛𝑡𝜇,𝑤𝜇;𝑞_∞d𝛽^(𝑎,𝑏)(𝑥)=𝑎^𝑛𝑎𝑤,𝑏𝑤,𝑎𝑡,𝑏𝑡;𝑞_∞𝑎𝑏𝑡𝑤/𝑞;𝑞_∞³𝜙₂𝑞^−𝑛,𝑞(),𝑞𝑎𝑤()𝑞𝑎𝑡₂();𝑎𝑏𝑡𝑤,0𝑞,𝑞, (2.22) which is (2.13) after multiplying (𝑟/𝑠;𝑞)_𝑛𝑠^𝑛/(𝑞;𝑞)_𝑛 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 max{|𝑥𝑞/𝑎|,|𝑢𝑞/𝑏|}<1. One has ^∞_𝑘=0(−1)^{𝑚+𝑛+𝑘}𝑞^{−2−𝑚2𝑛+𝑘−𝑚𝑘}𝑈_𝑛+𝑘𝑥,𝑦,𝑎∣𝑞𝑈_𝑚+𝑘𝑧𝑢,𝑣,𝑏∣𝑞^𝑘𝑞;𝑞_𝑘=𝑦𝑞/𝑎,𝑣𝑞/𝑏,𝑎𝑏𝑧;𝑞_∞𝑎^𝑛𝑏^𝑚𝑥𝑞/𝑎,𝑢𝑞/𝑏;𝑞_∞^∞_𝑘=0𝑦/𝑥,𝑞/(𝑎𝑏𝑧);𝑞_𝑘𝑞,𝑦𝑞/𝑎;𝑞_𝑘𝑥𝑏𝑧𝑞^𝑛𝑘×₂𝜙₁𝑣𝑢,𝑞^1+𝑘(𝑎𝑏𝑧)𝑣𝑞𝑏;𝑞,𝑎𝑢𝑧𝑞^𝑚+𝑘, (3.1) where 𝑦/𝑥=𝑞^−𝑓 and 𝑣/𝑢=𝑞^−𝑔 for nonnegative integers 𝑓,𝑔 and |𝑎𝑢𝑧𝑞^{−(𝑚+𝑘)}|<1.

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

Theorem 3.2. Assume that max{|𝑦𝑞/𝑎|,|𝑣𝑞/𝑏|}<1. One has ^∞_𝑘=0𝑉_𝑛+𝑘𝑥,𝑦,𝑎∣𝑞𝑉_𝑚+𝑘𝑢,𝑣,𝑏∣𝑞𝑧𝑞^𝑘𝑞;𝑞_𝑘=𝑥/𝑎,𝑢/𝑏;𝑞_∞𝑎^𝑛𝑏^𝑚𝑦/𝑎,𝑎𝑏𝑧,𝑣/𝑏;𝑞_∞^∞_𝑘=0𝑥/𝑦,𝑎𝑏𝑧;𝑞_𝑘𝑞^(1+𝑛)𝑘𝑞,𝑎𝑞/𝑦;𝑞_𝑘×₂𝜙₁𝑢𝑣,𝑎𝑏𝑧𝑞^𝑘𝑏𝑞𝑣;𝑞,𝑞^1+𝑚, (3.2) where 𝑥/𝑦=𝑞^−𝑓 and 𝑢/𝑣=𝑞^−𝑔 for nonnegative integers 𝑓,𝑔.

Corollary 3.3 (see [5, Theorem 4.5]). For 𝑓∈ℕ, one has ^∞_𝑘=0(−1)^𝑘𝑞^−𝑘2𝑈_𝑘𝑥,𝑦,𝑎∣𝑞𝑈_𝑘𝑢,𝑣,𝑏∣𝑞𝑧𝑞^𝑘𝑞;𝑞_𝑘=𝑎𝑏𝑧𝑞,𝑎𝑣𝑧𝑞,𝑦𝑞/𝑎;𝑞_∞𝑎𝑢𝑧𝑞,𝑥𝑞/𝑎;𝑞_∞³𝜙₂𝑦𝑥,1(),1𝑎𝑏𝑧()1𝑎𝑣𝑧(),𝑎𝑢𝑧𝑦𝑞𝑎;𝑞,𝑥𝑏𝑣𝑧𝑞𝑢, (3.3) provided that 𝑦/𝑥=𝑞^−𝑓 or 𝑣/𝑢=𝑞^−𝑓 and max{|𝑎𝑢𝑧𝑞^−𝑓|,|𝑥𝑞/𝑎|,|𝑥𝑏𝑣𝑧/𝑢|}<1.

Corollary 3.4. For 𝑓∈ℕ, one has ^∞_𝑘=0𝑉_𝑘𝑥,𝑦,𝑎∣𝑞𝑉_𝑘𝑧𝑢,𝑣,𝑏∣𝑞^𝑘𝑞;𝑞_𝑘=𝑎𝑢𝑧,𝑥/𝑎;𝑞_∞𝑎𝑏𝑧,𝑎𝑣𝑧,𝑦/𝑎;𝑞_∞³𝜙₂𝑥𝑎𝑏𝑧,𝑦,𝑎𝑣𝑧𝑎𝑢𝑧,𝑎𝑞𝑦;𝑞,𝑞, (3.4) provided that 𝑥/𝑦=𝑞^−𝑓 or 𝑢/𝑣=𝑞^−𝑓 and max{|𝑎𝑏𝑧|,|𝑎𝑣𝑧|,|𝑦/𝑎|}<1.

Remark 3.5. For 𝑚=𝑛=0, formula (3.1) and (3.2) reduce to (3.3) and (3.4), respectively, by 𝑞-Gauss sum [23, equation (II.8)], ₂𝜙₁𝑐;𝑐𝑎,𝑏𝑞,=𝑎𝑏𝑐/𝑎,𝑐/𝑏;𝑞_∞𝑐,𝑐/(𝑎𝑏);𝑞_∞ (3.5) and 𝑞-Chu-Vandermonde formula (2.2).

Proof of Theorems 3.1 and 3.2. The left-hand side of (3.1) is equal to ^∞_𝑘=0(−1)^{𝑚+𝑛+𝑘}𝑞^{−2−𝑚2𝑛+𝑘−𝑚𝑘}(−1)^𝑛+𝑘𝑞^2𝑛+𝑘𝑞𝑥/𝑦,𝑞𝑥/𝑎;𝑞_∞^∞_−∞𝜇^𝑛+𝑘𝑞𝑥𝜇(𝑎𝑦);𝑞_∞d𝛽^(𝑎,𝑦)𝜇×(−1)^𝑚+𝑘𝑞^2𝑚+𝑘𝑞𝑢/𝑣,𝑞𝑢/𝑏;𝑞_∞^∞_−∞𝑤^𝑚+𝑘𝑞𝑢𝑤(𝑏𝑣);𝑞_∞d𝛽^(𝑏,𝑣)(𝑤)𝑧^𝑘𝑞;𝑞_𝑘=1𝑞𝑥/𝑦,𝑞𝑥/𝑎,𝑞𝑢/𝑣,𝑞𝑢/𝑏;𝑞_∞^∞_−∞𝜇^𝑛𝑤^𝑚𝑞𝑥𝜇,𝑎𝑦𝑞𝑢𝑤(𝑏𝑣),𝜇𝑤𝑧;𝑞_∞d𝛽^(𝑎,𝑦)𝜇d𝛽^(𝑏,𝑣)(𝑤)=1𝑞𝑢/𝑣,𝑞𝑢/𝑏;𝑞_∞^∞_−∞1𝑞𝑥/𝑦,𝑞𝑥/𝑎;𝑞_∞^∞_−∞𝜇^𝑛𝑞𝑥𝜇𝑎𝑦,𝜇𝑤𝑧;𝑞_∞d𝛽^(𝑎,𝑦)𝜇×𝑤^𝑚𝑞𝑢𝑤(𝑏𝑣);𝑞_∞d𝛽^(𝑏,𝑣)(𝑤)=1𝑞𝑢/𝑣,𝑞𝑢/𝑏;𝑞_∞^∞_−∞𝑎^𝑛𝑎𝑤𝑧,𝑦𝑤𝑧;𝑞_∞𝑥𝑤𝑧;𝑞_∞³𝜙₂𝑞^−𝑛,𝑞(),𝑦𝑎𝑤𝑧𝑥𝑞();()𝑥𝑤𝑧,0𝑞,𝑞by2.8×𝑤^𝑚𝑞𝑢𝑤(𝑏𝑣);𝑞_∞d𝛽^(𝑏,𝑣)(𝑤)=1𝑞𝑢/𝑣,𝑞𝑢/𝑏;𝑞_∞^∞_−∞𝑎^𝑛𝑎𝑤𝑧,𝑦𝑤𝑧;𝑞_∞𝑥𝑤𝑧;𝑞_∞𝑞𝑦/𝑎,𝑥𝑤𝑧;𝑞_∞𝑞𝑥/𝑎,𝑦𝑤𝑧;𝑞_∞×₂𝜙₁𝑦𝑥,𝑞(𝑎𝑤𝑧)𝑦𝑞𝑎;𝑞,𝑥𝑤𝑧𝑞^𝑛×𝑤^𝑚𝑞𝑢𝑤(𝑏𝑣);𝑞_∞d𝛽^(𝑏,𝑣)(𝑤)=𝑎^𝑛𝑞𝑦/𝑎;𝑞_∞𝑞𝑢/𝑣,𝑞𝑢/𝑏,𝑞𝑥/𝑎;𝑞_∞^∞_𝑘=0𝑦/𝑥;𝑞_𝑘(−1)^𝑘𝑞^2𝑘+1𝑞,𝑦𝑞/𝑎;𝑞_𝑘𝑥𝑎𝑞^𝑛𝑘×^∞_−∞𝑤^𝑚𝑞𝑢𝑤(𝑏𝑣),𝑎𝑤𝑧𝑞^−𝑘;𝑞_∞d𝛽^(𝑏,𝑣)(𝑤)()=𝑎by2.22_𝑛𝑞𝑦/𝑎;𝑞_∞𝑞𝑢/𝑣,𝑞𝑢/𝑏,𝑞𝑥/𝑎;𝑞_∞^∞_𝑘=0𝑦/𝑥;𝑞_𝑘(−1)^𝑘𝑞^2𝑘+1𝑞,𝑦𝑞/𝑎;𝑞_𝑘𝑥𝑎𝑞^𝑛𝑘×𝑏^𝑚𝑞𝑢/𝑏,𝑞𝑢/𝑣,𝑎𝑏𝑧𝑞^−𝑘,𝑎𝑣𝑧𝑞^−𝑘;𝑞_∞𝑎𝑢𝑧𝑞^−𝑘;𝑞_∞×₃𝜙₂𝑞^−𝑚,𝑞^1+𝑘(),𝑣𝑎𝑏𝑧𝑢𝑞^1+𝑘();()=𝑎𝑎𝑢𝑧,0𝑞,𝑞by2.8_𝑛𝑞𝑦/𝑎;𝑞_∞𝑞𝑢/𝑣,𝑞𝑢/𝑏,𝑞𝑥/𝑎;𝑞_∞^∞_𝑘=0𝑦/𝑥;𝑞_𝑘(−1)^𝑘𝑞^2𝑘+1𝑞,𝑦𝑞/𝑎;𝑞_𝑘𝑥𝑎𝑞^𝑛𝑘×𝑏^𝑚𝑞𝑢/𝑏,𝑞𝑢/𝑣,𝑎𝑏𝑧𝑞^−𝑘,𝑎𝑣𝑧𝑞^−𝑘;𝑞_∞𝑎𝑢𝑧𝑞^−𝑘;𝑞_∞𝑞𝑣/𝑏,𝑎𝑢𝑧𝑞^−𝑘;𝑞_∞𝑢𝑞/𝑏,𝑎𝑣𝑧𝑞^−𝑘;𝑞_∞²𝜙₁𝑣𝑢,𝑞^1+𝑘(𝑎𝑏𝑧)𝑞𝑣𝑏;𝑞,𝑎𝑢𝑧𝑞^𝑚+𝑘, (3.6) which is the right hand side of (3.1) after simplification. Similarly, the right hand side of (3.2) is equivalent to ^∞_𝑘=0𝑥𝑦,𝑥𝑎;𝑞_∞^∞_−∞𝜇^𝑛+𝑘𝑥𝜇/𝑎𝑦;𝑞_∞d𝛼^(𝑎,𝑦)𝜇𝑢𝑣,𝑢𝑏;𝑞_∞×^∞_−∞𝑤^𝑚+𝑘𝑢𝑤/(𝑏𝑣);𝑞_∞d𝛼^(𝑏,𝑣)(𝑤)𝑧𝑞^𝑘𝑞;𝑞_𝑘=𝑢𝑣,𝑢𝑏;𝑞_∞^∞_−∞𝑥𝑦,𝑥𝑎;𝑞_∞^∞_−∞𝜇^𝑛𝑥𝜇/𝑎𝑦,𝜇𝑤𝑧;𝑞_∞d𝛼^(𝑎,𝑦)𝜇×𝑤^𝑚𝑢𝑤/(𝑏𝑣);𝑞_∞d𝛼^(𝑏,𝑣)(𝑤)=𝑢𝑣,𝑢𝑏;𝑞_∞^∞_−∞𝑎^𝑛𝑥𝑤𝑧;𝑞_∞𝑎𝑤𝑧,𝑦𝑤𝑧;𝑞_∞³𝜙₁𝑞^−𝑛𝑥,𝑎𝑤𝑧,𝑦;𝑥𝑤𝑧𝑞,𝑦𝑞^𝑛𝑎×𝑤^𝑚𝑢𝑤/(𝑏𝑣);𝑞_∞d𝛼^(𝑏,𝑣)(𝑤)()=𝑢by2.9𝑣,𝑢𝑏;𝑞_∞^∞_−∞𝑎^𝑛𝑥𝑤𝑧;𝑞_∞𝑎𝑤𝑧,𝑦𝑤𝑧;𝑞_∞𝑥/𝑎,𝑦𝑤𝑧;𝑞_∞𝑥𝑤𝑧,𝑦/𝑎;𝑞_∞²𝜙₁𝑥𝑦,𝑎𝑤𝑧𝑎𝑞𝑦;𝑞,𝑞^1+𝑛×𝑤^𝑚𝑢𝑤/(𝑏𝑣);𝑞_∞d𝛼^(𝑏,𝑣)(𝑤)=𝑎^𝑛𝑢/𝑣,𝑢/𝑏,𝑥/𝑎;𝑞_∞𝑦/𝑎;𝑞_∞^∞_𝑘=0𝑥/𝑦;𝑞_𝑘𝑞^(1+𝑛)𝑘𝑞,𝑎𝑞/𝑦;𝑞_𝑘×^∞_−∞𝑤^𝑚𝑢𝑤/(𝑏𝑣),𝑎𝑤𝑧𝑞^𝑘;𝑞_∞d𝛼^(𝑏,𝑣)(𝑤)()=𝑎by2.19_𝑛𝑢/𝑣,𝑢/𝑏,𝑥/𝑎;𝑞_∞𝑦/𝑎;𝑞_∞^∞_𝑘=0𝑥/𝑦;𝑞_𝑘𝑞^(1+𝑛)𝑘𝑞,𝑎𝑞/y;𝑞_𝑘𝑏^𝑚𝑎𝑢𝑧𝑞^𝑘;𝑞_∞𝑢/𝑣,𝑢/𝑏,𝑎𝑏𝑧𝑞^𝑘,𝑎𝑣𝑧𝑞^𝑘;𝑞_∞×₃𝜙₁𝑞^−𝑚,𝑎𝑏𝑧𝑞^𝑘,𝑢𝑣𝑎𝑢𝑧𝑞^𝑘;𝑞,𝑣𝑞^𝑚𝑏, (3.7) 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 (2.14)]). For 𝑠∈ℕ, one has ^∞_𝑛=0ℎ_𝑛+𝑠𝑥∣𝑞ℎ_𝑛𝑦∣𝑞ℎ_𝑛𝑤𝑧∣𝑞^𝑛𝑞;𝑞_𝑛=𝑦𝑧𝑤²;𝑞_∞𝑥,𝑤,𝑦𝑤,𝑧𝑤,𝑦𝑧𝑤;𝑞_∞⁶𝜙₅𝑞𝑤,𝑦𝑤,𝑧𝑤,𝑦𝑧𝑤,0,0𝑥,𝑤√𝑦𝑧,−𝑤√𝑦𝑧,𝑤√𝑦𝑧𝑞,−𝑤√;𝑦𝑧𝑞𝑞,𝑞^1+𝑠+𝑥^𝑠𝑦𝑧𝑥²𝑤²;𝑞_∞1/𝑥,𝑥𝑤,𝑥𝑦𝑤,𝑥𝑧𝑤,𝑥𝑦𝑧𝑤;𝑞_∞×₆𝜙₅√𝑥𝑤,𝑥𝑦𝑤,𝑥𝑧𝑤,𝑥𝑦𝑧𝑤,0,0𝑥𝑞,𝑥𝑤𝑦𝑧,−𝑥𝑤√𝑦𝑧,𝑥𝑤√𝑦𝑧𝑞,−𝑥𝑤√;𝑦𝑧𝑞𝑞,𝑞^1+𝑠. (4.1)

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

Theorem 4.2. For 𝑘∈ℕ, one has ^∞_𝑛=0𝑉_𝑛𝑥,𝑦,𝑎∣𝑞𝑉_𝑛𝑢,𝑣,𝑏∣𝑞𝑉_𝑛+𝑘𝑤𝑟,𝑠,𝑐∣𝑞^𝑛𝑞;𝑞_𝑛=𝑟/𝑠,𝑎𝑢𝑤𝑠,𝑏𝑥𝑤𝑠;𝑞_∞𝑠^𝑘𝑐/𝑠,𝑎𝑏𝑤𝑠,𝑎𝑣𝑤𝑠,𝑏𝑦𝑤𝑠;𝑞_∞^∞_𝑗=0𝑟/𝑐,𝑎𝑏𝑤𝑠,𝑎𝑣𝑤𝑠,𝑏𝑦𝑤𝑠;𝑞_𝑗𝑞^(1+𝑘)𝑗𝑞,𝑞𝑠/𝑐,𝑎𝑢𝑤𝑠,𝑏𝑥𝑤𝑠;𝑞_𝑗×₃𝜙₂𝑥𝑦,𝑢𝑣,𝑎𝑏𝑤𝑠𝑞^𝑗𝑎𝑢𝑤𝑠𝑞^𝑗,𝑏𝑥𝑤𝑠𝑞^𝑗;𝑞,𝑦𝑣𝑤𝑠𝑞^𝑗+𝑟/𝑐,𝑎𝑢𝑤𝑐,𝑏𝑥𝑤𝑐;𝑞_∞𝑐^𝑘𝑠/𝑐,𝑎𝑏𝑤𝑐,𝑎𝑣𝑤𝑐,𝑏𝑦𝑤𝑐;𝑞_∞×^∞_𝑗=0𝑟/𝑠,𝑎𝑏𝑤𝑐,𝑎𝑣𝑤𝑐,𝑏𝑦𝑤𝑐;𝑞_∞𝑞^(1+𝑘)𝑗𝑞,𝑞𝑐/𝑠,𝑎𝑢𝑤𝑐,𝑏𝑥𝑤𝑐;𝑞_𝑗³𝜙₂𝑥𝑦,𝑢𝑣,𝑎𝑏𝑤𝑐𝑞^𝑗𝑎𝑢𝑤𝑐𝑞^𝑗,𝑏𝑥𝑤𝑐𝑞^𝑗;𝑞,𝑦𝑣𝑤𝑐𝑞^𝑗, (4.2) 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 (4.13)] trilinear generating functions for 𝑞-Hermite polynomials 𝐻_𝑛(cos𝜃∣𝑞), 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 𝑟𝑠,𝑟𝑐;𝑞_∞^∞_𝑛=0^∞_−∞𝜇^𝑛+𝑘𝑟𝜇/(𝑐𝑠);𝑞_∞d𝛼^(𝑐,𝑠)𝜇𝑉_𝑛𝑥,𝑦,𝑎∣𝑞𝑉_𝑛𝑤𝑢,𝑣,𝑏∣𝑞^𝑛𝑞;𝑞_𝑛=𝑟𝑠,𝑟𝑐;𝑞_∞^∞_−∞𝜇^𝑘𝑟𝜇/(𝑐𝑠);𝑞_∞^∞_𝑛=0𝑉_𝑛𝑥,𝑦,𝑎∣𝑞𝑉_𝑛𝑢,𝑣,𝑏∣𝑞𝑤𝜇^𝑛𝑞;𝑞_𝑛d𝛼^(𝑐,𝑠)𝜇()=𝑟by3.4𝑠,𝑟𝑐;𝑞_∞^∞_−∞𝜇^𝑘𝑎𝑢𝑤𝜇,𝑥/𝑎;𝑞_∞𝑟𝜇/(𝑐𝑠),𝑎𝑏𝑤𝜇,𝑎𝑣𝑤𝜇,𝑦/𝑎;𝑞_∞×₃𝜙₂𝑥𝑎𝑏𝑤𝜇,𝑦,𝑎𝑣𝑤𝜇𝑎𝑢𝑤𝜇,𝑎𝑞𝑦;𝑞,𝑞d𝛼^(𝑐,𝑠)𝜇=𝑟𝑠,𝑟𝑐;𝑞_∞^∞_−∞𝜇^𝑘𝑎𝑢𝑤𝜇,𝑥/𝑎;𝑞_∞𝑟𝜇/(𝑐𝑠),𝑎𝑏𝑤𝜇,𝑎𝑣𝑤𝜇,𝑦/𝑎;𝑞_∞×𝑏𝑥𝑤𝜇,𝑦/𝑎;𝑞_∞𝑏𝑦𝑤𝜇,𝑥/𝑎;𝑞_∞³𝜙₂𝑥𝑦,𝑢𝑣;,𝑎𝑏𝑤𝜇𝑎𝑢𝑤𝜇,𝑏𝑥𝑤𝜇𝑞,𝑦𝑣𝑤𝜇d𝛼^(𝑐,𝑠)𝜇()=𝑟by1.9𝑠,𝑟𝑐;𝑞_∞^∞_𝑗=0𝑞^𝑗𝑐/𝑠;𝑞_∞𝑞,𝑞𝑠/𝑐;𝑞_𝑗𝑠𝑞^𝑗𝑘𝑎𝑢𝑤𝑠𝑞^𝑗,𝑏𝑥𝑤𝑠𝑞^𝑗;𝑞_∞𝑟𝑞^𝑗/𝑐,𝑎𝑏𝑤𝑠𝑞^𝑗,𝑎𝑣𝑤𝑠𝑞^𝑗,𝑏𝑦𝑤𝑠𝑞^𝑗;𝑞_∞×₃𝜙₂𝑥𝑦,𝑢𝑣,𝑎𝑏𝑤𝑠𝑞^𝑗𝑎𝑢𝑤𝑠𝑞^𝑗,𝑏𝑥𝑤𝑠𝑞^𝑗;𝑞,𝑦𝑣𝑤𝑠𝑞^𝑗+𝑟𝑠,𝑟𝑐;𝑞_∞^∞_𝑗=0𝑞^𝑗𝑠/𝑐;𝑞_∞𝑞,𝑞𝑐/𝑠;𝑞_𝑗×𝑐𝑞^𝑗𝑘𝑎𝑢𝑤𝑐𝑞^𝑗,𝑏𝑥𝑤𝑐𝑞^𝑗;𝑞_∞𝑟𝑞^𝑗/𝑠,𝑎𝑏𝑤𝑐𝑞^𝑗,𝑎𝑣𝑤𝑐𝑞^𝑗,𝑏𝑦𝑤𝑐𝑞^𝑗;𝑞_∞³𝜙₂𝑥𝑦,𝑢𝑣,𝑎𝑏𝑤𝑐𝑞^𝑗𝑎𝑢𝑤𝑐𝑞^𝑗,𝑏𝑥𝑤𝑐𝑞^𝑗;𝑞,𝑦𝑣𝑤𝑐𝑞^𝑗, (4.3) which is the right hand of (4.2) after simplification. The proof is complete.

Proof of Proposition 4.1. Noting the fact that 𝑉_𝑛(0,𝑦,1∣𝑞)=ℎ_𝑛(𝑦∣𝑞) and 𝑣𝑤²𝑦;𝑞_2𝑗=𝑤√𝑣𝑦,−𝑤√𝑣𝑦,𝑤√𝑣𝑦𝑞,−𝑤√𝑣𝑦𝑞;𝑞_𝑗,𝑠²𝑣𝑤²𝑦;𝑞_2𝑗=𝑠𝑤√𝑣𝑦,−𝑠𝑤√𝑣𝑦,𝑠𝑤√𝑣𝑦𝑞,−𝑠𝑤√𝑣𝑦𝑞;𝑞_𝑗. (4.4) If letting 𝑥=𝑢=𝑟=0 and 𝑎=𝑏=𝑐=1 in (4.2), the right hand side of (4.2) is equal to 𝑠^𝑘𝑠²𝑣𝑤²𝑦;𝑞_∞1/𝑠,𝑤𝑠,𝑣𝑤𝑠,𝑦𝑤𝑠,𝑦𝑣𝑤𝑠;𝑞∞^∞_𝑗=0𝑤𝑠,𝑣𝑤𝑠,𝑦𝑤𝑠,𝑦𝑣𝑤𝑠;𝑞_𝑗𝑞^(1+𝑘)𝑗𝑞,𝑞𝑠;𝑞_𝑗𝑠²𝑣𝑤²𝑦;𝑞_2𝑗+𝑣𝑤²𝑦;𝑞_∞𝑠,𝑤,𝑣𝑤,𝑦𝑤,𝑦𝑣𝑤;𝑞_∞^∞_𝑗=0𝑤,𝑣𝑤,𝑦𝑤,𝑦𝑣𝑤;𝑞_𝑗𝑞,𝑞/𝑠;𝑞_𝑗𝑣𝑤²𝑦;𝑞_2𝑗, (4.5) which reduces to the right hand side of (4.1) after setting (𝑠,𝑣)=(𝑥,𝑧). 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 (5.14)]). One has ^∞_{𝑚1,…,𝑚𝑘=0}ℎ_{𝑚1+⋯+𝑚𝑘}𝑎∣𝑞ℎ_𝑚1𝑡₁𝑡₂∣𝑞⋯ℎ_𝑚𝑘𝑡_2𝑘−1𝑡_2𝑘∣𝑞^𝑘_𝑗=1𝑡^𝑚𝑗_2𝑗𝑞;𝑞_𝑚𝑗=_2𝑘𝜙_2𝑘−1^{𝑡1,…,𝑡2𝑘𝑞/𝑎,0,…,0};^𝑞,𝑞𝑎;𝑞_∞∏^2𝑘_𝑗=1𝑡_𝑗;𝑞_∞+_2𝑘𝜙_2𝑘−1^{𝑎𝑡1,…,𝑎𝑡2𝑘𝑎𝑞,0,…,0};^𝑞,𝑞1/𝑎;𝑞_∞∏^2𝑘_𝑗=1𝑎𝑡_𝑗;𝑞_∞. (5.1)

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

Theorem 5.2. For 𝑠∈ℕ, one has ^∞_{𝑚1,…,𝑛𝑘=0}𝑉_{𝑚1+⋯+𝑚𝑘+𝑠}𝑦,𝑥,𝑎∣𝑞𝑉_𝑚1𝑦₁,𝑥₁,𝑎₁∣𝑞⋯𝑉_𝑚𝑘𝑦_𝑘,𝑥_𝑘,𝑎_𝑘𝑡∣𝑞^𝑚11⋯𝑡^𝑚𝑘_𝑘𝑞;𝑞_𝑚1⋯𝑞;𝑞_𝑚𝑘=𝑎^𝑠𝑦/𝑎,𝑦₁𝑡₁𝑎,…,𝑦_𝑘𝑡_𝑘𝑎;𝑞_∞𝑥/𝑎,𝑎₁𝑡₁𝑎,𝑥₁𝑡₁𝑎,…,𝑎_𝑘𝑡_𝑘𝑎,𝑥_𝑘𝑡_𝑘𝑎;𝑞_∞×_2𝑘+1𝜙_2𝑘𝑦𝑥,𝑎₁𝑡₁𝑎,𝑥₁𝑡₁𝑎,…,𝑎_𝑘𝑡_𝑘𝑎,𝑥_𝑘𝑡_𝑘𝑎𝑎𝑞𝑥,𝑦₁𝑡₁𝑎,…,𝑦_𝑘𝑡_𝑘𝑎;𝑞,𝑞^1+𝑠+𝑥^𝑠𝑦/𝑥,𝑦₁𝑡₁𝑥,…,𝑦_𝑘𝑡_𝑘𝑥;𝑞_∞𝑎/𝑥,𝑎₁𝑡₁𝑥,𝑥₁𝑡₁𝑥,…,𝑎_𝑘𝑡_𝑘𝑥,𝑥_𝑘𝑡_𝑘𝑥;𝑞_∞×_2𝑘+1𝜙_2𝑘𝑦𝑎,𝑎₁𝑡₁𝑥,𝑥₁𝑡₁𝑥,…,𝑎_𝑘𝑡_𝑘𝑥,𝑥_𝑘𝑡_𝑘𝑥𝑥𝑞𝑎,𝑦₁𝑡₁𝑥,…,𝑦_𝑘𝑡_𝑘𝑥;𝑞,𝑞^1+𝑠, (5.2) provided that the right hand side of (5.2) is convergent.

Corollary 5.3 (see [20, equation (1.12)]). For 𝑠∈ℕ, one has ^∞_{𝑛1,…,𝑛𝑘=0}ℎ_{𝑛1+⋯+𝑛𝑘+𝑠}𝑥∣𝑞ℎ_𝑛1𝑥₁∣𝑞…ℎ_𝑛𝑘𝑥_𝑘𝑡∣𝑞^𝑛11𝑞;𝑞_𝑛1⋯𝑡^𝑛𝑘_𝑘𝑞;𝑞_𝑛𝑘=1𝑥,𝑡₁,𝑥₁𝑡₁,…,𝑡_𝑘,𝑥_𝑘𝑡_𝑘;𝑞_∞^2𝑘𝜙_2𝑘−1𝑡₁,𝑥₁𝑡₁,…,𝑡_𝑘,𝑥_𝑘𝑡_𝑘𝑞𝑥;,0,…,0𝑞,𝑞^1+𝑠+𝑥^𝑠1/𝑥,𝑥𝑡₁,𝑥𝑥₁𝑡₁,…,𝑥𝑡_𝑘,𝑥𝑥_𝑘𝑡_𝑘;𝑞_∞^2𝑘𝜙_2𝑘−1𝑥𝑡₁,𝑥𝑥₁𝑡₁,…,𝑥𝑡_𝑘,𝑥𝑥_𝑘𝑡_𝑘;𝑞𝑥,0,…,0𝑞,𝑞^1+𝑠. (5.3)

Remark 5.4. Letting 𝑦=𝑦₁=⋯=𝑦_𝑘=𝑠=0, 𝑎=𝑎₁=⋯=𝑎_𝑘=1 and taking (𝑥,𝑥_𝑖,𝑡_𝑖)=(𝑎,𝑡_𝑖/𝑡_𝑖+1,𝑡_2𝑖) in Theorem 5.2, formula (5.2) reduces to (5.1). Setting 𝑦=𝑦₁=⋯=𝑦_𝑘=0 and 𝑎=𝑎₁=⋯=𝑎_𝑘=1 in Theorem 5.2, formula (5.2) reduces to (5.3).

Proof. The left hand side of (5.2) is equal to 𝑦𝑥,𝑦𝑎;𝑞_∞^∞∞_{−∞𝑚1,…,𝑛𝑘=0}𝑉_𝑚1𝑦₁,𝑥₁,𝑎₁∣𝑞⋯𝑉_𝑚𝑘𝑦_𝑘,𝑥_𝑘,𝑎_𝑘×𝑡∣𝑞₁𝜇^𝑚1⋯𝑡_𝑘𝜇^𝑚𝑘𝑞;𝑞_𝑚1⋯𝑞;𝑞_𝑚𝑘𝜇^𝑠d𝛼^(𝑎,𝑥)𝜇𝑦𝜇/(𝑎𝑥);𝑞_∞=𝑦𝑥,𝑦𝑎;𝑞_∞^∞_−∞𝜇^𝑠𝑦₁𝑡₁𝜇,…,𝑦_𝑘𝑡_𝑘𝜇;𝑞_∞𝑎₁𝑡₁𝜇,𝑥₁𝑡₁𝜇,…,𝑎_𝑘𝑡_𝑘𝜇,𝑥_𝑘𝑡_𝑘𝜇,𝑦𝜇/(𝑎𝑥);𝑞_∞d𝛼^(𝑎,𝑥)𝜇=𝑦𝑥,𝑦𝑎;𝑞_∞^∞_𝑟=0𝑞^𝑟𝑥/𝑎;𝑞_∞𝑞,𝑞𝑎/𝑥;𝑞_𝑟×𝑎𝑞^𝑟𝑠𝑦₁𝑡₁𝑎𝑞^𝑟,…,𝑦_𝑘𝑡_𝑘𝑎𝑞^𝑟;𝑞_∞𝑎₁𝑡₁𝑎𝑞^𝑟,𝑥₁𝑡₁𝑎𝑞^𝑟,…,𝑎_𝑘𝑡_𝑘𝑎𝑞^𝑟,𝑥_𝑘𝑡_𝑘𝑎𝑞^𝑟,𝑦𝑞^𝑟/𝑥;𝑞_∞+^∞_𝑟=0𝑞^𝑟𝑎/𝑥;𝑞_∞𝑞,𝑞𝑥/𝑎;𝑞_𝑟×𝑥𝑞^𝑟𝑠𝑦₁𝑡₁𝑥𝑞^𝑟,…,𝑦_𝑘𝑡_𝑘𝑥𝑞^𝑟;𝑞_∞𝑎₁𝑡₁𝑥𝑞^𝑟,𝑥₁𝑡₁𝑥𝑞^𝑟,…,𝑎_𝑘𝑡_𝑘𝑥𝑞^𝑟,𝑥_𝑘𝑡_𝑘𝑥𝑞^𝑟,𝑦𝑞^𝑟/𝑎;𝑞_∞=𝑎^𝑠𝑦/𝑎,𝑦₁𝑡₁𝑎,…,𝑦_𝑘𝑡_𝑘𝑎;𝑞_∞𝑥/𝑎,𝑎₁𝑡₁𝑎,𝑥₁𝑡₁𝑎,…,𝑎_𝑘𝑡_𝑘𝑎,𝑥_𝑘𝑡_𝑘𝑎;𝑞_∞×_2𝑘+1𝜙_2𝑘𝑦𝑥,𝑎₁𝑡₁𝑎,𝑥₁𝑡₁𝑎,…,𝑎_𝑘𝑡_𝑘𝑎,𝑥_𝑘𝑡_𝑘𝑎𝑎𝑞𝑥,𝑦₁𝑡₁𝑎,…,𝑦_𝑘𝑡_𝑘𝑎;𝑞,𝑞^1+𝑠+𝑥^𝑠𝑦/𝑥,𝑦₁𝑡₁𝑥,…,𝑦_𝑘𝑡_𝑘𝑥;𝑞_∞𝑎/𝑥,𝑎₁𝑡₁𝑥,𝑥₁𝑡₁𝑥,…,𝑎_𝑘𝑡_𝑘𝑥,𝑥_𝑘𝑡_𝑘𝑥;𝑞_∞,×_2𝑘+1𝜙_2𝑘𝑦𝑎,𝑎₁𝑡₁𝑥,𝑥₁𝑡₁𝑥,…,𝑎_𝑘𝑡_𝑘𝑥,𝑥_𝑘𝑡_𝑘𝑥𝑥𝑞𝑎,𝑦₁𝑡₁𝑥,…,𝑦_𝑘𝑡_𝑘𝑥;𝑞,𝑞^1+𝑠, (5.4) 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 𝑛∈ℕ, one has ^𝑛_𝑘=0𝑈_𝑘𝑐𝑞^𝑘−1(),𝑑,𝑏∣𝑞−1^𝑘𝑞^{𝑘2𝑛𝑘−}𝑞^−𝑛,𝑎;𝑞_𝑘𝑞,𝑐/𝑑,𝑐/𝑏;𝑞_𝑘𝑐𝑎𝑏𝑑^𝑘=𝑐/(𝑎𝑑);𝑞_𝑛𝑐/𝑑,𝑐/𝑏;𝑞_𝑛²𝜙₁𝑞^−𝑛,𝑑𝑞^1−𝑛𝑐𝑎𝑑𝑞^1−𝑛𝑐;𝑞,𝑐𝑞^𝑛𝑏. (6.1)

Theorem 6.2. For 𝑛∈ℕ, one has ^𝑛_𝑘=0𝑉_𝑛−𝑘𝑎𝑞^𝑘𝑞,𝑑,𝑏∣𝑞^−𝑛,𝑎/𝑑,𝑎/𝑏;𝑞_𝑘𝑞,𝑐;𝑞_𝑘𝑏𝑐𝑑𝑞^𝑛𝑎^𝑘=𝑐𝑑/𝑎;𝑞_𝑛𝑏^𝑛𝑐;𝑞_𝑛²𝜙₁𝑞^−𝑛,𝑎𝑑𝑎𝑞^1−𝑛();𝑐𝑑𝑞,𝑎𝑞.𝑏𝑐 (6.2)

Corollary 6.3. For 𝑛∈ℕ, one has ^𝑛_𝑘=0𝑔_𝑘𝑞𝑑∣𝑞^−𝑛;𝑞_𝑘𝑞;𝑞_𝑘𝑐𝑞^𝑛𝑑^𝑘=𝑐𝑑;𝑞_𝑛1𝜙₁𝑞^−𝑛𝑑𝑞^1−𝑛𝑐;𝑞,𝑑𝑞_𝑛𝑘=0,ℎ_𝑛−𝑘𝑞𝑑∣𝑞^−𝑛;𝑞_𝑘𝑞;𝑞_𝑘𝑐𝑑𝑞^𝑛𝑘=𝑐𝑑;𝑞_𝑛2𝜙₁𝑞^−𝑛𝑞,0_1−𝑛();𝑞𝑐𝑑𝑞,𝑐. (6.3)

Remark 6.4. Letting 𝑏→∞ in Theorems 6.1 and 6.2, noting the fact that lim_𝑏→∞(−1)^𝑘𝑞^−𝑘2𝑏^−𝑘𝑈_𝑘𝑐𝑞^𝑘−1,𝑑,𝑏∣𝑞=lim_𝑏→∞𝑏^𝑘−𝑛𝑉_𝑛−𝑘𝑎𝑞^𝑘,𝑑,𝑏∣𝑞=1. (6.4) Formula (6.1) and (6.2) reduce to 𝑞-Chu-Vandermonde formula (2.2), respectively. Replacing 𝑐 by 𝑎𝑐 and letting (𝑎,𝑏)=(0,1) 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 𝑐 by 𝑐𝜇/(𝑏𝑑) on both sides of the second formula in (2.2), we have ^𝑛_𝑘=0𝑞^−𝑛,𝑎;𝑞_𝑘𝑞;𝑞_𝑘𝑐𝑞^𝑛𝑎𝑏𝑑^𝑘⋅𝜇^𝑘𝑐𝜇𝑞^𝑘(𝑏𝑑);𝑞_∞=𝑐𝜇/(𝑎𝑏𝑑),𝑐𝜇𝑞^𝑛/(𝑏𝑑);𝑞_∞𝑐𝜇𝑞^𝑛/(𝑏𝑑);𝑞_∞. (6.5) Applying moments ∫^∞_−∞d𝛽^(𝑏,𝑑)(𝜇) in (6.5), using (2.10) and (2.13), we gain ^𝑛_𝑘=0𝑞^−𝑛,𝑎;𝑞_𝑘𝑞;𝑞_𝑘𝑐𝑞^𝑛𝑎𝑏𝑑^𝑘(−1)^𝑘𝑞^−𝑘2𝑐𝑞^𝑘𝑑,𝑐𝑞^𝑘𝑏;𝑞_∞𝑈_𝑘𝑐𝑞^𝑘−1=(),𝑑,𝑏∣𝑞𝑐/𝑎𝑑,𝑐𝑞^𝑛/𝑑,𝑐𝑞^𝑛/𝑏;𝑞_∞𝑐𝑞^𝑛/(𝑎𝑑);𝑞_∞²𝜙₁𝑞^−𝑛,𝑑𝑞^1−𝑛𝑐𝑎𝑑𝑞^1−𝑛𝑐;𝑞,𝑐𝑞^𝑛𝑏, (6.6) which is (6.1) after simplification. Similarly, setting 𝑎 by 𝑎𝑤/(𝑏𝑑) on both sides of the second formula in (2.2), we obtain ^𝑛_𝑘=0𝑞^−𝑛;𝑞_𝑘𝑞,𝑐;𝑞_𝑘𝑏𝑐𝑑𝑞^𝑛𝑎^𝑘⋅𝑤^𝑛−𝑘𝑎𝑤𝑞^𝑘/(𝑏𝑑);𝑞_∞=(−1)^𝑛𝑞^𝑛2(𝑏𝑐𝑑/𝑎)^𝑛𝑐;𝑞_𝑛⋅𝑎𝑤𝑞^1−𝑛/(𝑏𝑐𝑑);𝑞_∞𝑎𝑤𝑞/(𝑏𝑐𝑑),𝑎𝑤/(𝑏𝑑);𝑞_∞. (6.7) Utilizing moments ∫^∞_−∞d𝛼^(𝑏,𝑑)(𝑤) in (6.7), by (2.11) and (2.12), we have ^𝑛_𝑘=0𝑞^−𝑛;𝑞_𝑘𝑞,𝑐;𝑞_𝑘𝑏𝑐𝑑𝑞^𝑛𝑎^𝑘1𝑎𝑞^𝑘/𝑑,𝑎𝑞^𝑘/𝑏;𝑞_∞𝑉_𝑛−𝑘𝑎𝑞^𝑘=(),𝑑,𝑏∣𝑞−1^𝑛𝑞^𝑛2(𝑏𝑐𝑑/𝑎)^𝑛𝑐;𝑞_𝑛𝑎𝑞^1−𝑛/(𝑐𝑑);𝑞_∞𝑎𝑞/(𝑐𝑑),𝑎/𝑑,𝑎/𝑏;𝑞_∞²𝜙₁𝑞^−𝑛,𝑎𝑑𝑎𝑞^1−𝑛();𝑐𝑑𝑞,𝑎𝑞,𝑏𝑐 (6.8) which becomes (6.2) after simplification. The proof is complete.

Acknowledgments

The author would like to express deep appreciation to the referee and the editor for their helpful suggestions.