q-Difference equations for the 2-iterated q-Appell and mixed type q-Appell polynomials

In this article, the authors establish the recurrence relations and q-difference equations for the 2-iterated q-Appell polynomials. The recurrence relations and the q-difference equations for the 2-iterated q-Bernoulli polynomials, the q-Euler polynomials and the q-Genocchi polynomials are also derived. An analogous study of certain mixed type q-special polynomials is also presented.

is due to the fact that q-series has popped in such diverse areas as statistical mechanics, quantum groups, transcendental number theory, etc.
The definitions and notations of q-calculus reviewed here are taken from [3].
The q-analogue of the shifted factorial (a) n is defined by (a; q) 0 = 1, (a; q) n = n−1 m=0 (1 − q m a), n ∈ N. (1.1) The q-analogues of a complex number a and of factorial function are defined by [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 defined by The q-exponential functions are defined as: The q-derivative D q f of a function f at a point 0 = z ∈ C is defined as: Also, for any two arbitrary functions f (z) and g(z), the following relation for the q-derivative holds true: D q,z ( f (z)g(z)) = f (z)D q,z g(z) + g(qz)D q,z f (z). (1.7) Al-Salaam [1] introduced the family of q-Appell polynomials {A n,q (x)} n≥0 and studied some of its properties. The n-degree polynomials A n,q (x) are called q-Appell provided they satisfy the following q-differential equation: D q,x {A n,q (x)} = [n] q A n−1,q (x), n = 0, 1, 2, . . . ; q ∈ C; 0 < q < 1. (1.8) The q-Appell polynomials A n,q (x) are also defined by means of the following generating function [1]: where A q (t) := ∞ n=0 A n,q t n [n] q ! , A 0,q = 1; A q (t) = 0. (1.10) It is to be noted that A q (t) is an analytic function at t = 0 and A n,q := A n,q (0) (1.11) are the q-Appell numbers. Based on appropriate selection for the function A q (t), different members belonging to the family of q-Appell polynomials can be obtained. These members are mentioned in Table 1.  q-Euler polynomials and number [8,19] 2 eq (t)+1 q-Genocchi polynomials and number [11,19] 2t The q-Appell polynomials are the generalizations of the Appell polynomials A n (x) [4] which are determined by the power series expansion of the product A(t)e xt , that is The function A(t)e xt is called generating function of the sequence of polynomials A n (x) and the function A(t) is an analytic function at t = 0 and A n := A n (0) (1.13) are the Appell numbers.
The set of all Appell sequences form an abelian group under the umbral composition of polynomial sequences. The Appell polynomial sequences are well studied from different aspects [4][5][6][7]10,25] due to their applications in various fields. One aspect of such study is to find recurrence relations and differential equations for the Appell sequences. For example, He and Ricci [10] established the finite order recurrence relations and differential equations for the Appell sequences using factorization method.
Recently, certain mixed special polynomial families related to the Appell sequences are studied in a systematic way, see for example, [13,15,18,24,32]. These polynomials are studied thoroughly due to their applications in various fields of mathematics, physics and engineering. The properties of these mixed special families lie within the properties of the parent polynomials. To find the differential, integro-differential and partial differential equations for a mixed special polynomial family [18] is a recent investigation [32]. The recurrence relations, differential equations and other results of these mixed type special polynomials can be used to solve the existing as well as new emerging problems in certain branches of science. Introducing a determinant form for the mixed special polynomials via operational and algebraic techniques is a new study, which has been taken into consideration and can be helpful for computation purposes. The technique of combining two sequences by means of umbral composition [27] is a systematic way of constructing mixed special sequences.
Khan and Raza [14] introduced and studied a composite family by combining two different sets of Appell sequences namely the 2-iterated Appell polynomial sequences A [2] n (x), which are defined by means of the following generating relation: A [2] n (x) t n n! . (1.14) The set of all 2-iterated Appell sequences A [2] n (x) also form an abelian group under the operation of umbral composition. With the help of determinant form of the 2-iterated Appell sequences A [2] n (x) considered in [17], it may be possible to compute the coefficients or the value in a chosen point, for particular sequences of the 2-iterated Appell polynomial family, through an efficient and stable Gaussian algorithm. It can also be useful in finding the solution of general linear interpolation problem. Khan and Riyasat [16] studied the differential and integral equations for the 2-iterated Appell polynomial sequences A [2] n (x) and mentioned that the respective differential equations can be used to study the d-orthogonality property for these sequences, thus making these 2-iterated sequences important from different view point.
In 1985, Roman proposed an approach similar to the umbral approach under the area of nonclassical umbral calculus which is called q-umbral calculus [26,28]. By using q-analysis and q-umbral calculus, the q-polynomials are introduced and characterized by several authors, for this see [8,9,11,[29][30][31].
The 2-iterated q-Appell polynomials (2IqAP) are introduced and studied by combining two different sets of q-Appell polynomials using the concept of q-umbral composition of polynomial sequences. The generating function for the 2-iterated q-special polynomial families is introduced using a different approach based on replacement techniques. The 2IqAP are defined by means of the following generating function [17]: respectively. It is to be noted that A I q (t) and A II q (t) are analytic functions at t = 0 and A [2] n,q := A [2] n,q (0) are the 2-iterated q-Appell numbers.
The series definition for the 2IqAP A [2] n,q (x) is given as: denotes the 2-iterated q-Appell numbers. We recall that the set of all q-Appell sequences is closed under the operation of q-umbral composition of polynomial sequences. Under this operation the set of all q-Appell sequences is an abelian group and it can be seen by considering the fact that every q-Appell sequence is of the form (1.20) and that umbral composition of q-Appell sequences corresponds to multiplication of these formal q-power series in the operator D q . In view of above fact, it is remarked that the 2-iterated q-Appell polynomials A [2] n,q (x) satisfy the following relation: Again, if p n,q (x) and q n,q (x) = n k=0 q n,k;q x k are sequences of q-polynomials, then the q-umbral composition of q n,q (x) with p n,q (x) is defined to be the sequence q n,q ( p q (x)) = n k=0 q n,k;q p k;q (x), (1.22) which is equivalent to condition (1.18).
Since the generating function of the 2IqAP is of the form A q (t)e q (xt), with A q (t) as the product of two similar functions of t. Therefore, the set of all 2IqAP sequences also form an abelian group under the operation of q-umbral composition. The determinant form of the 2IqAP introduced in [17] can also be used for computation purposes. That is by applying stable Gaussian algorithm, it may be possible to compute the coefficients or the value in a chosen point, for particular sequences of the 2IqAP family.
Since the generating function (1.15) of the 2IqAP sequences is the product of two functions A I q (t) and A II q (t), which shows that by making appropriate selection for the function A I q (t) and A II q (t), different members belonging to the family of the 2-iterated q-Appell polynomials can be obtained. By making the combinations of two same members of the q-Appell family in the 2-iterated q-Appell family, a new 2-iterated q-polynomial can be obtained. The generating function and series definition of these 2-iterated q-polynomials are given in Table 2.
By taking the combination of any two different members of the q-Appell family in the 2-iterated q-Appell family, a new mixed type q-special polynomial can be obtained. The generating function and series definition of these mixed type q-special polynomials are given in Table 3. Table 2 Certain members belonging to the 2-iterated q-Appell family Notation and name of the resultant 2IqAP Generating function Series definition G [2] n,q (x): Hence, the generating function (1.15) in its product form gains special importance due to the fact by making the combinations of some other generalized members belonging to the q-Appell family, certain other new 2iterated and mixed type q-special polynomials related to the 2IqAP family can be obtained. These generalized members are listed in Table 4.
The determinant forms related to the 2IqBP B [2] n,q (x), 2IqEP E [2] n,q (x), 2IqGP G [2] n,q (x), qBEP B E n,q (x), qBGP B G n,q (x) and qEGP E G n,q (x) are considered in [17]. The respective determinant forms can be useful in finding the solution of various general linear interpolation problems.
Also, the shapes of the 2IqBP B [2] n,q (x), 2IqEP E [2] n,q (x), 2IqGP G [2] n,q (x), qBEP B E n,q (x), qBGP B G n,q (x) and qEGP E G n,q (x) are displayed and the real and complex zeros of these polynomials are computed for index n = 1, 2, 3, 4 and q = 1/2 (0 < q < 1) using Matlab in [17]. The distribution and structure of the zeros is also displayed. By finding the real zeros of these polynomials, the approximate solutions and spectral properties of these mixed q-special polynomials are also studied using "Matlab". Thus, making these polynomials important from another view point.
In the 21st century, the computing environment is making more and more rapid progress. Using computer, a realistic study of these new mixed q-special numbers and polynomials seems very interesting. Further, by using numerical investigations and computer experiments, we can observe an interesting phenomenon of "scattering" of the zeros and the regular lattice behavior of almost all of the real and complex zeros of mixed type special and q-special polynomials for higher values of n, i.e. n > 4 or for a fixed range of n i.e. n = 1 − 50 and for different values of q or for a range of q such that 0 < q < 1. However, to this point there have been no such investigations for the mixed type special and q-special polynomials. Hence, this will gain extra importance to these new classes of mixed special polynomials.
Since the raising operators are not available for q-polynomials, although lowering operators always exist. Recently, Mahmudov [20] used the lowering operators to study the q-difference equations for the q-Appell polynomials A n,q (x). This provides motivation to establish the q-difference equations for the 2-iterated q-Appell and mixed type q-Appell polynomials.
The article is organized as follows. In Sect. 2, the recurrence relations and q-difference equations for the 2-iterated q-Appell polynomials are introduced. In Sect. 3, the recurrence relations and q-difference equations for the 2-iterated q-Bernoulli, 2-iterated q-Euler and 2-iterated q-Genocchi polynomials and certain mixed type q-special polynomials are also established.

Recurrence relations and q-difference equations
In this section, the recurrence relations and q-difference equations for the 2-iterated q-Appell polynomials are established. To derive the recurrence relation for the 2IqAP A [2] n,q (x), the following result is proved: Eqs. (1.16) and (1.17), assume that

Theorem 2.1 For two different sets of q-Appell polynomials A I n,q (x) and A II n,q (x) and with A I q (t) and A II q (t) defined by
respectively. Then, the following linear homogeneous recurrence relation for the 2-iterated q-Appell polynomials A [2] n,q (x) holds true: n − k s q β n−k−s γ s A [2] k,q (x) + xq n A [2] n−1,q (x), n ≥ 1.
(2.10) Therefore, considering the lowering operator as: it follows that which is the k-times derivative operator for the 2IqAP A [2] n,q (x). Replacement of x by qx in generating function (1.15) and then differentiation of the resultant equation with respect to t using formula (1.7), gives Further, use of formula (1.7) in Eq. (2.13) and then multiplication by t yields [n] q A [2] n,q (qx) which on simplifying and interchanging the sides becomes ∞ n=0 [n] q A [2] n,q (qx) [n] q A [2] n,q (qx) [n] q A [2] n,q (qx) Next, the q-difference equation for the 2IqAP A [2] n,q (x) is derived by proving the following result: [2] n,q (x) satisfy the following q-difference equation: [2] n,q (x) − [n] q A [2] n,q (qx) = 0.
In the next section, the recurrence relations and q-difference equations for the 2-iterated q-Appell polynomials given in Table 2 and for the mixed type q-special polynomials given in Table 3 are established.

Examples
To derive the recurrence relations and q-difference equations for the 2-iterated q-Bernoulli, 2-iterated q-Euler and 2-iterated q-Genocchi polynomials, the following examples are considered: t e q (t)−1 (that is when the 2IqAP A [2] n,q (x) reduce to the 2IqBP B [2] n,q (x)) in Eqs. respectively.
In view of generating functions (Table 1, I, II), the above equations give 2t e q (t)+1 (that is when the 2IqAP A [2] n,q (x) reduce to the qBGP B G n,q (x)) in Eqs. (2.1)-(2.3), respectively, so that respectively.
In view of generating functions (Table 1, I, III), the above equations give 2t e q (t)+1 (that is when the 2IqAP A [2] n,q (x) reduce to the qEGP E G n,q (x)) in Eqs. (2.1)-(2.3), respectively, so that respectively.
In view of generating functions (Table 1, II, III), the above equations give and  In the next section, further applications and importance of the 2-iterated and mixed type q-special polynomials are discussed.

Further applications
The orthogonal polynomials in general and the classical orthogonal polynomials in particular have been the object of extensive works. They are connected with numerous problems of applied mathematics, theoretical physics, chemistry, approximation theory and several other mathematical branches.
During the last 20 years, there has been a growing interest in multiple orthogonal polynomials. However, it is only recently that the examples of multiple orthogonal polynomials have appeared in the literature. A convenient framework to discuss such examples consists in considering a subclass of multiple orthogonal polynomials known as d-orthogonal polynomials [5,7]. The notion of d-dimensional orthogonality for polynomials [23] is the generalization of ordinary orthogonality for polynomials. The problem of finding all polynomial sequences, which are at the same time q-Appell polynomials and d-orthogonal is considered in [33].
The new investigations and important results related to the 2-iterated q-Appell and mixed type q-special polynomials are derived in [17] and are briefly discussed in Sect. 1, which make these polynomials important from different view points. The results which are derived in Sects. 2 and 3 also acquire special importance. This paper is a first attempt to establish the recurrence relations and q-difference equations for the mixed type q-special polynomials and can also be taken to solve various problems arising in different areas of science and engineering. These q-recurrence relations and q-difference equations of the 2-iterated q-Appell and mixed type q-special polynomials can be used to study the d-orthogonality property of these polynomials. This is obvious that when these polynomials become orthogonal, these can be useful to other fields such as in wavelet analysis. The series expansions and continuous wavelet transforms can be derived in terms of 2-iterated Appell, 2-iterated q-Appell and mixed type q-Appell polynomials and their particular members. This aspect may be considered in further investigation.