Densely Generated 2D q-Appell Polynomials of Bessel Type and q-addition Formulas

Mumtaz Riyasat abstract: The article aims to introduce a densely generated class of 2D q-Appell polynomials of Bessel type and to investigate their properties. It is advantageous to consider the 2D q-Bernoulli, 2D q-Roger Szegö and 2D q-Al-Salam Carlitz polynomials of Bessel type as their special members and to derive the q-determinant forms and certain q-addition formulas for these polynomials. The article concludes with a brief view on discrete q-Bessel convolution of the 2D q-Appell polynomials.


Introduction
The q-analogues appear in the diverse areas of combinatorics and fluid mechanics, quantum group theory, group representation theory, number theory, statistical mechanics, quantum mechanics and also have an intimate connection with commutativity relations and Lie algebra and q-deformed super algebras. We begin this section by describing notations and definitions of q-Calculus following from [3].
The q-analogues of the shifted factorial (a) n are given by (a; q) 0 = 1, (a; q) n = n−1 m=0 (1 − q m a) (a, q ∈ R; n ∈ N) (1.1) The q-analogues of a complex number a and of the factorial function are given by [n] q ! = n m=1 [m] q = [1] q [2] q · · · [n] q = (q; q) n (1 − q) n q = 1; n ∈ N; [0] q ! = 1; q ∈ C; 0 < q < 1 . (1. 3) The Gauss q-binomial coefficient n k q is given by = (q; q) n (q; q) k (q; q) n−k (k = 0, 1, . . . , n). (1.4) The q-analogue of the function (x + y) n is defined as: The q-analogues of exponential functions are given by Consequently, we note that Moreover, the functions e q (x) and E q (x) satisfy the following properties: where the q-derivative D q f of a function f at a point 0 = z ∈ C is defined as follows: For any two arbitrary functions f (z) and g(z), the following relation holds true: The special polynomials of two variables are important from the point of view of applications in different branches of pure and applied mathematics and physics. These polynomials allow the derivation of a number of useful identities and relations in a fairly straight forward way and help in introducing new families of special polynomials.
For q ∈ C, 0 < |q| < 1, the 2D q-Appell polynomials A n,q (x, y) are defined by the following generating function [7]: The function A q (t) is analytic at t = 0 and A n,q := A n,q (0, 0). It is to be noted that where A n,q (x) are the q-Appell polynomials [1].
We present the polynomials belonging to the 2D q-Appell family A n,q (x, y) (for appropriate choice of A q (t)) in Table 1.
Densely Generated 2D q-Appell Polynomials of Bessel Type and q-addition Formulas

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The importance of generalized Bessel functions stems from their wide use in applications. The scattering of free or weakly bounded electrons by intense laser fields is an example where generalized Bessel functions an important role. The analytical and numerical study of these functions has revealed their interesting properties, which led to their extension to two-dimensional domain. The relevance of these functions in mathematical physics has been emphasized, since they provide analytical solutions to partial differential equations such as the multi-dimensional diffusion equation, the Schrodinger and Klein-Gordon equation.
The two-dimensional Bessel functions are defined by the following generating function [6]: (1.14) Motivated by the importance and relevance of the two-dimensional Bessel functions, in 2018 Riyasat and Khan introduce the two-dimensional (or 2D) q-Bessel polynomials p n,q (x, y), which are defined by means of the following generating function: Taking y = 0 and x = 0, consecutively in above equation, the two forms of q-Bessel polynomials p n,q (x) and P n,q (x) are deduced, which are defined by the following generating functions: The q-Bessel polynomials p n,q (x) are the q-analogue of Carlitz type Bessel polynomials p n (x) [5].
In this article, the densely generated 2D q-Appell polynomials of Bessel type are defined by means of generating function and determinant form. The 2D q-Bernoulli, 2D q-Roger Szegö and 2D q-Al-Salam Carlitz polynomials of Bessel type are considered as their special members and corresponding determinant forms and q-addition formulas are derived.

2D q-Appell polynomials of Bessel type
We introduce a dense form of generating function for the 2D q-Appell polynomials of Bessel type. For this, we prove the following result: Theorem 2.1. The generating function for the 2D q-Appell polynomials of Bessel type is given by Proof. Expanding the exponential function e q (x(1 − √ 1 − 2t)) and then replacing the powers x 0 , . . . , x n by the sequences A 0,q (x, y), A 1,q (x, y), . . . , A n,q (x, y) in the l.h.s. and replacing x by A 1,q (x, y) in the r.h.s. of equation (1.16), if we sum up the terms in the l.h.s. of the resulting equation, we find that which, on using equation (1.11) in the l.h.s. and denoting the resulting polynomials in the r.h.s. by A p n,q (x, y) := p n,q (A 1,q (x, y)), yields the assertion (2.1).
• It is to be noted that for x = y = 0, we have • Again, for y = 0, we have the q-Appell polynomials of Bessel type defined by Theorem 2.2. The 2D q-Appell polynomials of Bessel type satisfy the following q-differential recurrence relations: Proof. Differentiating generating equation (2.1) with respect to x and y, respectively using equation (1.8) and then using formula: (2.8) in resultant equations and after rearranging the summations, we get recurrence relations (2.6) and (2.7).
Next, we find the determinant form for the 2D q-Appell polynomials of Bessel type by proving the following theorem: Theorem 2.3. The following determinant form for the 2D q-Appell polynomials of Bessel type holds true: where the coefficients b −1 i,j;q are given by (2.10) Densely Generated 2D q-Appell Polynomials of Bessel Type and q-addition Formulas

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Proof. We recall the following determinant form for the q-Bessel polynomial sequences {p n,q (x)} n∈N from [10]: where the coefficients b −1 i,j;q are given by equation (2.10).
Now, replacing the powers x m by the q-polynomials A m,q (x, y) for m = 0, 1, . . . , n, n + 1 in the r.h.s. and replacing 1 x by A 1,q 1 x , 1 y in the l.h.s. of equation (2.11) and then using relation: in the l.h.s. of resultant equation, yields assertion (2.9).
We now consider the following interesting remarks, which may give several important polynomials and corresponding results related to the 2D q-Appell polynomials of Bessel type.

Remark 2.4. For the choice of
we have the 2D q-Bernoulli polynomials of Bessel type (2DqBPoBT) defined by 14) The determinant form for the 2D q-Bernoulli polynomials of Bessel type is given by (2.15)

Remark 2.5. For the choice of
we have the 2D q-Roger-Szegö polynomials of Bessel type (2DqRSPoBT) defined by

16)
which for x = y = 0 gives The determinant form for the 2D q-Roger-Szegö polynomials of Bessel type is given by (2.18)

Remark 2.6. For the choice of
we have the 2D q-Al-Salam Carlitz polynomials of Bessel type (2DqACPoBT) defined by

19)
which for x = y = 0 gives n,q t n [n] q ! . (2.20) The determinant form for the 2D q-Al-Salam Carlitz polynomials of Bessel type is given by In the next section, we establish the q-addition formulas for the members belonging to the 2D q-Appell polynomials of Bessel type.