A note on generalized $q$-difference equations for general Al-Salam--Carlitz polynomials

In this paper, we deduce the generalized $q$-difference equations for general Al-Salam--Carlitz polynomials and generalize Arjika's recently results [$q$-difference equation for homogeneous $q$-difference operators and their applications, J. Differ. Equ. Appl. {\bf 26}, 987--999 (2020)]. In addition, we obtain transformational identities by the method of $q$-difference equation. Moreover, we deduce $U(n+1)$ type generating functions and Ramanujan's integrals involving general Al-Salam--Carlitz polynomials by $q$-difference equation.


Introduction
In this paper, we refer to the general references [1] for definitions and notations. Throughout this paper, we suppose that 0 < q < 1. For complex numbers a, the q-shifted factorials are defined by: and (a 1 , a 2 , ..., a m ; q) n = (a 1 ; q) n (a 2 ; q) n ...(a m ; q) n , where m is a positive integer and n is a non-negative integer or ∞.
We remark in passing that, in a recently-published survey-cum-expository review article, the so-called (p, q)calculus was exposed to be a rather trivial and inconsequential variation of the classical q-calculus, the additional parameter p being redundant or superfluous (see, for details, ( [7], p. 340)).
The basic (or q-) series and basic (or q-) polynomials, especially the basic (or q-) hypergeometric functions and basic (or q-) hypergeometric polynomials, are applicable particularly in several diverse areas [see also ([3], pp. 350-351)]. In particular, the celebrated Chu-Vandermonde summation theorem and its known q-extensions, which have already been demonstrated to be useful (see, for details, [1,8,9,10]).
The usual q-differential operator, or q-derivative, is defined by [11,12,13] The Leibniz rule for D a and θ a are the following identities [11,12,14] D n a { f (a)g(a)} = The following property of D q is straightforward and important [15]  They play important roles in the theory of q-orthogonal polynomials. In fact, there are two families of these polynomials: one with continuous orthogonality and another with discrete orthogonality, which given explicitly in the book of Koekoek-Swarttouw-Lesky [17,Eqs. (14.24.1) and (14.25.1)]. For further information about q-polynomials, see [18,19,20,17,21,22].
The generalized Al-Salam-Carlitz polynomials [23, Eq. (4.7)] whose generating functions are [23,Eqs. (4.10) and (4.11))] c; q; xt , |xt| < 1. (1.9) Chen and Liu [11,12] gave the clever way of parameter augmentation by use of the following two qexponential operators which is rich and powerful tool for basic hypergeometric series, especially makes many famous results easily fall into this framework. For further information about q-exponential operators, see [11,12,24,25,26,27]. Recently, Srivastava, Arjika and Sherif Kelil [28] introduced the following homogeneous q-difference operator E(a, b; D q ) by The operators (1.11) have turned out to be suitable for dealing with a generalized Cauchy polynomials p n (x, y, a) For more information about the relations between operators and q-polynomials, see [28]. Liu [15,29] deduced several results involving Bailey's 6 ψ 6 , q-Mehler formulas for Rogers-Szegö polynomials and q-integral of Sears' transformation by the following q-difference equations. Arjika [30] continue to consider the following generalized q-difference equations. ). Let f (a, x, y) be a three-variable analytic function in a neighbourhood of (a, x, y) = (0, 0, 0) ∈ C 3 . If f (a, x, y) satisfied the q-difference equation

Proposition 1 ([29, Theorems 1 and 2]). Let f (a, b) be a two-variable analytic function in a neighbourhood of
In this paper, our goal is to generalize the results of Arjika [30] in section 2. We first construct the following q-operators We remark that the q-operator (1.20) is a particular case of the homogeneous q-difference operator T(a, b, cD x ) (see [45]) by taking a = (a, b, c), b = (d, e) and c = y. We also built the relations between operators T(a, b, c, d, e, yD x ), E(a, b, c, d, e, yθ x ) and the new generalized The paper is organized as follows: In section 2, we state two theorems and give the proofs. In section 3, we gain generalize generating functions for new generalized Al-Salam-Carlitz polynomials by using the method of q-difference equations perspectively. In section 4, we obtain a transformational identities involving generating functions for generalized Al-Salam-Carlitz polynomials by q-difference equations. In section 5, we deduce U(n + 1) type generating functions for generalized Al-Salam-Carlitz polynomials by q-difference equation. In section 6, we deduce generalizations of Ramanujan's integrals.

Main results and proofs
In this section, we give the following two theorems.
(I) If f (a, b, c, d, e, x, y) satisfied the difference equation 3) reduces to (1.18). To determine if a given function is an analytic function in several complex variables, we often use the following Hartogs's theorem. For more information, please refer to [31,32].

Lemma 8 ([33, Hartogs's theorem]). If a complex-valued function is holomorphic (analytic) in each variable separately in an open domain D ∈ C n , then it is holomorphic (analytic) in D.
In order to prove Theorem 4, we need the following fundamental property of several complex variables.
Proof of Theorem 4. (I) From the Hartogs's theorem and the theory of several complex variables, we assume that On one hand, substituting (2.7) into (2.1) yields which is equal to Equating coefficients of y k on both sides of equation (2.9), we have 1 (a, b, c, d, e, xq)], (2.10) which is equivalent to 1 (a, b, c, d, e, x)}.
By iteration, we gain Proof of Theorem 6. From the theory of several complex variables, we begin to solve the q-difference. First we may assume that a, b, c, d, e, x)y k . (2.12) Substituting this equation into (2.12) and compare coefficients of y k (k ≥ 1), we readily find that 1 (a, b, c, d, e, xq)], (2.13) which equals 1 (a, b, c, d, e, x)}.

Generating functions for new generalized Al-Salam-Carlitz polynomials
In this section we generalized generating functions for the new generalized Al-Salam-Carlitz polynomials by the method of q-difference equations.
We start with the following lemmas.

Lemma 10 ([35]). The Cauchy polynomials as given below
together with the following Srivastava-Agarwal type generating function (see also [36]): So, f (a, b, c, d, e, x, y) is equal to equal to the right-hand side of equation (3.5). Similarly, by denoting the right-hand side of equation ( f (a, b, c, d, e, x, y) is equal to the right-hand side of equation ( n k (−1) j q n j−( j 2 ) (q −k , xt; q) j (xt) j , (3.11) and it is easy to check that (3.11) satisfies (2.1), so we have f (a, b, c, d, e, x, y n (x, y|q). (3.12) Setting y = 0 in (3.12), it becomes The proof of Theorem 18 is complete.
The proof of Theorem 20 is complete.

Transformational identities from q-difference equations
Liu [23] gave some important transformational identities by the method of q-difference operator. For more details, please refer to [17,23,38].
In this section we deduce the following transformational identities involving generating functions for new generalized Al-Salam-Carlitz polynomials by the method of q-difference equation.

Theorem 22. Let A(k) and B(k) satisfy
and we have   Using equation (4.2), we can deduce the Corollary 23.
5. U(n + 1) type generating functions for generalized Al-Salam-Carlitz polynomials Multiple basic hypergeometric series associated to the unitary U(n + 1) group have been investigated by various authors, see [39,40]. In [39], Milne initiated theory and application of the U(n + 1) generalization of the classical Bailey transform and Bailey lemma, which involves the following nonterminating U(n + 1) generalizations of the q-binomial theorem. In this section, we deduce U(n + 1) type generating functions for generalized Al-Salam-Carlitz polynomials by the methodn of q-difference equation.

Generalization of Ramanujan's integrals
The following integral of Ramanujan [41] are quite famous.
So we have which is equal to the left-hand side of equation (6.2). The proof of Theorem 29 is complete.

Concluding Remarks and Observations
In our present investigation, we have introduced a set of two q-operators T (a, b, c, d, e, yD x ) and E(a, b, c, d, e, yθ x ) with to applying them to generalize Arjika's recently results [30], and derive transformational identities by means of the q-difference equations. We have also derived U(n + 1)-type generating functions and Ramanujan's integrals involving general Al-Salam-Carlitz polynomials by means of the q-difference equations.
It is believed that the q-series and q-integral identities, which we have presented in this paper, as well as the various related recent works cited here, will provide encouragement and motivation for further researches on the topics that are dealt with and investigated in this paper.
In conclusion, we find it to be worthwhile to remark that some potential further applications of the methodology and findings, which we have presented here by means of the q-analysis and the q-calculus, can be found in the study of the zeta and q-zeta functions as well as their related functions of Analytic Number Theory (see, for example, [42,43]; see also [8]) and also in the study of analytic and univalent functions of Geometric Function Theory via number-theoretic entities (see, for example, [44]).