q-difference equations for the composite 2D q-Appell polynomials and their applications

Abstarct: he main aim of this article is to introduce a new class of composite 2D q-Appell polynomials and to study their properties. The generating function, series definition and some explicit relations for these polynomials are derived. These polynomials are studied from determinantal view point and their q-recurrence relations and q-difference equations are established. The composite 2D q-Bernoulli, q-Euler and q-Genocchi and composite q-Bernoulli–Euler, q-Bernoulli–Genocchi and q-Euler–Genocchi polynomials are studied as particular members of this class. Certain interesting examples are considered in terms of these members to give the applications of main results.


ABOUT THE AUTHOR
Mumtaz Riyasat is a postdoctoral fellow (NBHM) at the Department of Mathematics, Aligarh Muslim University, India. She received her PhD degree in Mathematics at AMU in the year 2016 and received the best Thesis award of the year 2016. She has published around 10 research paper in highly reputed international journals. Her research interests include special functions and operational techniques, q-series and qpolynomials, differential equations, fractional calculus, approximation theory.

PUBLIC INTEREST STATEMENT
The studies on introducing mixed type polynomials via operational techniques and establishing their determinantal forms, recurrence relations and differential equations via factorization method have been done (Araci, Riyasat, Wani, & Khan 2017;Khan & Riyasat, 2016a). The q-series and q-polynomials have many applications in different fields of mathematics, physics and engineering. Recently, a new replacement technique has been adopted to introduce mixed type q-special polynomials and a different method is used to establish their q-recurrence relations and q-difference equations (Khan & Riyasat, 2016b;Srivastava, Khan, & Riyasat, in press). Motivated by this, in this article, we introduce a composite class of 2D q-Appell polynomials and studied its several properties. The generating function, series definition, explicit relations, determinantal definition, q-recurrence relations and q-difference equations are established.
Certain interesting examples are framed in terms of the composite 2D q-Bernoulli, q-Euler and q-Genocchi and composite q-Bernoulli-Euler, q-Bernoulli-Genocchi and q-Euler-Genocchi polynomials to give the applications of main results. This process can be used to establish further quite a wide variety of formulas for several other q-special polynomials and can be extended to derive new relations for conventional and generalized q-polynomials.

Introduction and preliminaries
The subject of q-calculus started appearing in the nineteenth century due to its applications in various fields of mathematics, physics and engineering. Recently, it seems to have more usefulness in combinatorics and fluid mechanics, quantum mechanics, having an intimate connection with commutativity relations and Lie algebra (Bohner & Ünal, 2005;Ernst, 2017;Floreanini & Vinet, 1992, 1993a, 1993bKatriel, 1998;Miller, 1970). The definitions and notations of q-calculus reviewed here are taken from Andrews, Askey, and Roy (1999).
The q-analogue of the shifted factorial (a) n is defined by The q-analogues of a complex number a and of the factorial function are defined by The q-binomial coefficient n k q is defined by The q-analogue of the function (x + y) n is defined as: The exponential functions are defined as: The functions e q (x) and E q (x) satisfy the following properties: The q-derivative D q f of a function f at a point 0 ≠ z ∈ ℂ is defined as: For any two arbitrary functions f(z) and g(z), the q-derivative satisfy (1.1) (a; q) 0 = 1, (a; q) n = n−1 ∏ m=0 (1 − q m a), n ∈ ℕ. (1.2) (1 − q) n , q ≠ 1; n ∈ ℕ, [0] q ! = 1, q ∈ ℂ; 0 < q < 1.

III.
q-Genocchi polynomials (Araci et al., 2014;Mahmudov, 2013) Different members belonging to the family of the 2D q-Appell polynomials A n, q (x, y) can be obtained by choosing appropriate A q (t). These polynomials are mentioned in Table 2.
The q-Bernoulli, q-Euler and q-Genocchi numbers have deep connections with number theory and occur in combinatorics. We give first few values of q-Bernoulli numbers B n, q (Ernst, 2006), q-Euler numbers E n, q (Ernst, 2006) and q-Genocchi numbers G n,q Araci, Acikgoz, Jolany, and He (2014) in Table 3, which will be used later.
For decades, various families of q-polynomials have been investigated rather widely and extensively due to their potentially usefulness in such wide variety of fields as theory of partitions, number theory, combinatorial analysis, finite vector spaces, Lie theory, etc. for this see Ernst (2017), Bohner and Ünal (2005), Floreanini and Vinet (1992), Floreanini and Vinet (1993a), Floreanini and Vinet (1993b), Katriel (1998), Miller (1970. The study of differential and difference equations is a wide field in pure and applied mathematics, physics and engineering. The problems arising in different areas of science and engineering are usually expressed in terms of differential or difference equations, which in most of the cases have special functions as their solutions. During the past three decades, the development of non-linear analysis, dynamical systems and their applications to science and engineering has stimulated renewed enthusiasm for the theory of differential or difference equations. The difference equations are important as they deal with the discrete and differential equations deal with the continuous and both our mathematical and physical universes are inherently discrete. The prominence of the role of the stability properties makes the difference among the numerical analysis and other branches of mathematics which also use the difference equations as a main tool. For example, in combinatorics, difference equations are very important. Differential equations play an important role in modelling virtually every physical, technical or biological process, from celestial motion, to bridge design, to interactions between neurons. Many fundamental laws of physics and chemistry can be formulated as differential equations. In biology and economics, differential equations are used to model the behaviour of complex systems (Johnson, 1913).  Notes: It is to be noted that the q-Genocchi polynomials G n, q (x) (Table 1 (III)) do not fulfil all requirements of q-Appell sequences as for instance the degree of G n, q (x) is n − 1 which can be seen from Table 3 that G 0, q = 0 ⇒ G 0, q (x) = 0, however the degree of all other q-Appell polynomials is n. Therefore, G n, q (x) is considered in the class of polynomial sequences which are not q-Appell in the strong sense.
The article is organized as follows. In Section 2, a composite class of the 2D q-Appell polynomials is introduced by means of generating function, series definition and determinantal definition. The q-recurrence relations and q-difference equations for these polynomials are established. In Section 3, certain examples of the members of the composite 2D q-Appell polynomials are considered as applications.

Composite 2D q-Appell polynomials
To introduce the Composite 2D q-Appell polynomials (C2DqAP), we consider two different sets of the 2D q-Appell polynomials such that where and respectively.
In order to give the generating function for the composite 2D q-Appell polynomials, we prove the following result: Theorem 2.1 The composite 2D q-Appell polynomials A A n, q (x, y) are defined by the following generating function: Proof In view of Equations (1.5)-(1.7), (2.1) is written as Expanding the summation and then replacing the powers 1, (x + y) 1 q , … , (x + y) n−1 q , (x + y) n q by the polynomials 1, A II 1, q (x, y), … , A II n−1, q (x, y), A II n, q (x, y) in the l.h.s. and (x + 0) 1 by the polynomial A II 1, q (x, 0) in the r.h.s. of above equation, we have which on summing up the series and using Equation (2.2) in the l.h.s. and denoting the resultant C2DqAP in the r.h.s. by A A n, q (x, y) yields assertion (2.5).
Remark 2.1 It is remarked that, for y = 0, the composite 2D q-Appell polynomials A A n, q (x, y) reduce to the composite q-Appell polynomials (Khan & Riyasat, 2016), such that It is to be noted that Next, we give the series definition for the composite 2D q-Appell polynomials A A n, q (x, y) by proving the following result: Theorem 2.2 The composite 2D q-Appell polynomials A A n, q (x, y) are defined by the following series definition: Proof Using Equations (2.1) and (2.4) in generating function (2.5) and then using Cauchy product rule in the l.h.s. of the resultant equation, we have Equating the coefficients of same powers of t in both sides of Equation (2.11), assertion (2.10) follows.

Theorem 2.3 The composite 2D q-Appell polynomials A A n, q (x, y) satisfy the following relations:
Proof Using Equation (1.5) in the l.h.s. of generating function (2.5) gives which on applying the Cauchy product rule in the r.h.s. and then comparing the coefficients of same powers of t in both sides of resultant equation yields assertion (2.12).
Using Equation (1.7) in the l.h.s.generating function (2.5) gives which on applying the Cauchy product rule in the r.h.s. and then comparing the coefficients of same powers of t in both sides of resultant equation yields assertion (2.13).
Using Equation (1.6) in the l.h.s. of generating function (2.5) gives respectively.
The k-times lowering operators are given by Proof Differentiating generating function (2.5) k-times with respect to x and y and using the fact that it follows that which on using Cauchy product rule and then equating the coefficients of same powers of t in both sides of resultant equations yields assertions (2.20) and (2.21).
Since the following operational relations holds: which gives the lowering operators as: Therefore, in view of Equations (2.20)-(2.21) and (2.30)-(2.31), the k-times lowering operators for the C2DqAP A A n, q (x, y) are given by Equations (2.22) and (2.23).
To derive the q-recurrence relations for the composite 2D q-Appell polynomials, the following result is proved: Theorem 2.5 For two different sets of 2D q-Appell polynomials A I n, q (x, y) and A II n, q (x, y) and with A I q (t) and A II q (t) defined by Equations (2.3) and (2.4), assume that and respectively. Then, the following linear homogeneous recurrence relations for the composite 2D q-Appell polynomials A A n, q (x, y) holds true for n ≥ 1: and Proof Taking x → qx and then q-derivative with respect to t on both sides of generating function (2.5) using formula (1.10) gives which on multiplying by t on both sides and then simplifying the resultant equation yields (2.38) (2.39) Using Equations (2.32)-(2.35) with generating function (2.5) gives which on rearranging the summations using Cauchy product rule in the r.h.s. becomes The above equation can also be written as which on using the fact that 0 = 0 = 0 and then equating the coefficients of same powers of t in both sides of the above equation yields assertion (2.36).
Solving the summation for k = 1 in the first term of the r.h.s. of Equation (2.36) yields assertion (2.37).
Replacing k by n − k + 1 in Equation ( (2.44) In Section 3, certain examples are considered as applications of the results derived above.

Applications
The generating function, series definition, q-recurrence relations and q-difference equations of some members of the C2DqAP A A n, q (x, y) are derived by considering the following examples: By making suitable selection for the functions A I q (t) and A II q (t), the members belonging to the family of composite 2D q-Appell polynomials A A n, q (x, y) can be obtained. The generating functions and other results for these polynomials are given in Table 4.
Using these values in Theorems 2.6 and 2.7, we conclude the following: To show the graphical representation of the C2DqBP B B n, q (x, y) for an index n = 4 and q = 1 2 (0 < q < 1), we require the first few expressions of the composite q-Bernoulli polynomials B B n, q (x) (Khan & Riyasat, 2016). These expressions are given in Table 5.

S.No. A I q (t); A II q (t) Name of the resultant C2DqAP and related number Generating function, series definition and other relations
Using the expressions of first five B B n, 1∕2 (x) from Table 5 in Equation (Table 4 (I)), we find With the help of Matlab and using above expression, the surface plot for the C2DqBP is drawn, for this see Figure 1.
ing these values in Theorems 2.6 and 2.7, we conclude the following: Corollary 3.3 The following linear homogeneous recurrence relation for the composite 2D q-Euler polynomials E E n, q (x, y) holds true for n ≥ 1: Corollary 3.4 The composite 2D q-Euler polynomials E E n, q (x, y) satisfy the following q-difference equations: To show the graphical representation of the C2DqEP E E n, q (x, y) for an index n = 4 and q = 1 2 (0 < q < 1), we require the first few expressions of the composite q-Euler polynomials E E n, q (x) (Khan & Riyasat, 2016). These expressions are given in Table 6.
Using the expressions of first five E E n, 1∕2 (x) from Table 6 in Equation (Table 4 (II)), we find With the help of Matlab and using above expression, the surface plot for the C2DqEP is drawn, for this see Figure 2. (3.6) The determinantal definition for the C2DqEP E E n, q (x, y) can be obtained by taking 0, q = 1, i, q = 1 2 (i = 1, 2, … , n) (for which determinantal definition of A n, q (x) reduce to E n, q (x)) and A n, q (x, y) = E n, q (x, y) (n = 0, 1, 2, … , n) in determinantal definition (2.18) of the C2DqAP A A n, q (x, y).
2t e q (t)+1 , the C2DqAP A A n, q (x, y) reduce to the C2DqGP G G n, q (x, y), therefore in view of Equations (2.32)-(2.35), we find Using these values in Theorems 2.6 and 2.7, we conclude the following: Corollary 3.5 The following linear homogeneous recurrence relation for the composite 2D q-Genocchi polynomials G G n, q (x, y) holds true for n ≥ 1: Corollary 3.6 The composite 2D q-Genocchi polynomials G G n, q (x, y) satisfy the following q-difference equations: To show the graphical representation of the C2DqGP G G n, q (x, y) for an index n = 4 and q = 1 2 (0 < q < 1), we require the first few expressions of the composite q-Genocchi polynomials G G n, q (x) (Khan & Riyasat, 2016). These expressions are given in Table 7.
Using the expressions of first five G G n, 1∕2 (x) from Table 7 in Equation (Table 4 (III)), we find (3.9) (3.10) Notes: From Table 7, it can be seen that degree of composite q-Genocchi polynomials is n − 2. Hence, the polynomials G G n, q (x) are considered in the class of polynomial sequences which are not composite q-Appell in the strong sense. With the help of Matlab and using above expression, the surface plot for the C2DqGP is drawn, for this see Figure 3.
Example 3.4 We know that for A I q (t) = t e q (t)−1 and A II q (t) = 2 e q (t)+1 , the C2DqAP A A n, q (x, y) reduce to the C2DqBEP B E n, q (x, y), therefore in view of Equations (2.32)-(2.35), we find Using these values in Theorems 2.6 and 2.7, we conclude the following: Corollary 3.7 The following linear homogeneous recurrence relation for the composite 2D q-Bernoulli-Euler polynomials B E n, q (x, y) holds true for n ≥ 1: Corollary 3.8 The composite 2D q-Bernoulli-Euler polynomials B E n, q (x, y) satisfy the following q-difference equations: (3.12) (3.13) (3.14) � q n (x − 1 2q To show the graphical representation of the C2DqBEP B E n, q (x, y) for an index n = 4 and q = 1 2 (0 < q < 1), we require the first few expressions of the composite q-Bernoulli-Euler polynomials B E n, q (x) (Khan & Riyasat, 2016). These expressions are given in Table 8.
Using the expressions of first five B E n, 1∕2 (x) from Table 8 in Equation (Table 4 (IV)), we find With the help of Matlab and using above expression, the surface plot for the C2DqBEP is drawn, for this see Figure 4.
Example 3.5 We know that for A I q (t) = t e q (t)−1 and A II q (t) = 2t e q (t)+1 , the C2DqAP A A n, q (x, y) reduce to the C2DqBGP B G n, q (x, y), therefore in view of Equations (2.32)-(2.35), we find n = −1 q B n, q ; 0 = 0; 1 = − 1 (3.17) To show the graphical representation of the C2DqBGP B G n, q (x, y) for an index n = 4 and q = 1 2 (0 < q < 1), we require the first few expressions of the composite q-Bernoulli-Genocchi polynomials B G n, q (x) (Khan & Riyasat, 2016). These expressions are given in Table 9.
Using the expressions of first five B G n, 1∕2 (x) from Table 9 in Equation (Table 4 (V)), we find With the help of Matlab and using above expression, the surface plot for the C2DqBGP is drawn, for this see Figure 5.
Example 3.6 We know that for A I q (t) = 2 e q (t)+1 and A II q (t) = 2t e q (t)+1 , the C2DqAP A A n, q (x, y) reduce to the C2DqEGP E G n, q (x, y), therefore in view of Equations (2.32)-(2.35), we find n = 1 2 E n−1, q ; 0 = 0; 1 = − 1 2 , n = 1 2q G n, q ; 0 = 1 q ; 1 = − 1 q , n = q−1 2 n ∑ k=0 � n k � q E k, q , n ≥ 1; 0 = q+1 2 and n = q−1 2q n ∑ k=0 � n k � q G k, q , n ≥ 1; 0 = 1 q . Using these values in Theorems 2.6 and 2.7, we conclude the following: Corollary 3.11 The following linear homogeneous recurrence relation for the composite 2D q-Euler-Genocchi polynomials E G n, q (x, y) holds true for n ≥ 1: Corollary 3.12 The composite 2D q-Euler-Genocchi polynomials E G n, q (x, y) satisfy the following q-difference equations: (3.21) Notes: From Table 9, it can be seen that degree of composite q-Bernoulli-Genocchi polynomials is n − 1. Hence, the polynomials B G n, q (x) are considered in the class of polynomial sequences which are not composite q-Appell in the strong sense. From Figures 1-6, we interpret that the polynomials B B n, q (x, y), E E n, q (x, y), G G n, q (x, y), B E n, q (x, y), B G n, q (x, y) and E G n, q (x, y) achieve the maximum value at x = y = −4 and minimum at x = 4 and y > −4.   Surface plot of G G 4,1/2 (x,y)  Surface plot of E G 4,1/2 (x,y)