A note on q-analogue of Hermite-poly-Bernoulli numbers and polynomials

Abstract. In this paper, we introduce the Hermite-based poly-Bernoulli numbers and polynomials with q-parameter and give some of their basic properties including not only addition property, but also derivative properties and integral representations. We also define the Hermitebased λ-Stirling polynomials of the second kind and then provide some relations, identities of these polynomials related to the Stirling numbers of the second kind. We derive some symmetric identities for these families of special functions by applying the generating functions.


Introduction
Throughout this paper, we use the following notations, N = {1, 2, 3, . . .} denotes the set of natural numbers, N 0 = {1, 2, 3, . . .} denotes the set of non negative integer, Z denotes the set of integers and C denotes the set of complex numbers respectively.
These polynomials are usually defined by the generating function: H n (x, y) t n n! , (1.1) As is well known, the Bernoulli polynomials are defined by the generating function as t e t − 1 e xt = ∞ n=0 B n (x) t n n! , (see   The classical polylogarithm function Li k (z) is z m m k , (k ∈ Z), (see [10,11,12]).
For k = 1, we have The Stirling number of the first kind is given by S 1 (n, l)x l , (n ≥ 0), (see [4,9,26] This paper organized as follows. In Section 2, we introduce a new class of Hermite poly-Bernoulli numbers and polynomials with q-parameter. In Section 3, we establish some identities of these polynomials. In Section 4, we derive some properties of the Stirling numbers of the second kind. In Section 5, we derive symmetric identities for these generalized polynomials by using different analytical means on their respective generating functions.

2.
A note on q-analogue of Hermite-poly-Bernoulli numbers and polynomials In this section, we define a q-analogue of Hermite-poly-Bernoulli numbers and polynomials and its properties. Definition 2.1. For n ≥ 0, n, k ∈ Z, 0 ≤ q < 1, we introduce a q-analogue of Hermite-poly-Bernoulli polynomials by means of the following generating function: where Li k,q (t) = ∞ n=0 t n [n] k q ! is the k-th q-polylogarithm function (see [4,6,23]).
n,q (0, 0) are called a q-analogue poly-Bernoulli numbers. Remark 2.1. For y = 0 in (2.1), the result reduces to the known result of Hwang et al. [9] as follows: Thus by (2.1) and (2.2), we get n (x, y) = H B n (x, y) (n ≥ 0), (see [21,22]). Theorem 2.1. (Addition Property), we have Equating the coefficients of t n n! in both sides, we get (2.3). Theorem 2.2. (Derivative Properties) Each of the following formula holds true: Proof. The proof follows from (2.1). So we omit them. n+1,q (r, y) n + 1 Proof. Using the derivative properties of H B (k) n,q (x, y) given in Theorem 2.2, we easily get the asserted results. So we omit them.
Theorem 2.4. The following formula holds true: which implies the desired result (2.4).

Main Results
In this section, we establish some properties of q-analogue of Hermitepoly-Bernoulli polynomials by using generating function.
In particular for k = 2, we have Replacing n by n − m in R.H.S. of above equation, we have On comparing the coefficients of t n n! on both sides of the above equation, we get the result (3.2).
Comparing the coefficients of t n n! on both sides, we get the result (3.3).
Equating the coefficients of t n n! in both sides, we get (3.4). n,q (x, y) holds true: Proof. We replace t by t + u and rewrite the generating function (2.1) as Replacing x by z in the above equation and equating the resulting equation to the above equation, we get On expanding exponential function, (3.3) gives which on using formula [27, p.52 (2)] Finally on equating the coefficients of the like powers of t and u in the above equation, we get the required result. n,q (z, y) holds true: p−n,q (x, y).

The q-analogue of Hermite-based Stirling polynomials of the second kind
In this section, we introduce q-analogue of Hermite-based Stirling polynomials of the second kind is defined by For x = y = 0 in (4.1), S 2 (n, m) = S 2 (n, m; 0, 0) are called the Stirling numbers of the second kind (see [4,9,26]). We give some relations and properties belonging to the Hermite-based Stirling polynomials of the second kind by the following consecutive Theorems.    Proof. From (2.1), we have Replacing n by n − a in above equation, we get Comparing the coefficients of t n n! in above equation, we get (4.2).

(4.3)
Proof. Replacing x by x + u in (2.1), we have On comparing the coefficients of t n n! in both sides, we get at the desired result (4.3).  Proof. Using the definition (2.1), we have On comparing the coefficients of t n n! on either side, we get the result (4.4). (−1) l+p+1 l!S 2 (p + 1, l) Proof. Equation (2.1) can be written as Replacing n by n − p in the R.H.S. of above equation and comparing the coefficient of t n n! on either side, we get the result (4.5). Proof. By using (2.1) and (4.1), we have On comparing the coefficients of t n n! in both sides, we arrive at the desired result (4.6).

Symmetric identities
In this section, we give general symmetry identities for the q-poly-Bernoulli polynomials B (k) n,q (x) and the Hermite poly-Bernoulli polynomials H B (k) n,q (x, y) with q parameter by applying the generating function (2.1) and (2.2). The results extend some known identities of Khan [11][12][13][14], Pathan and Khan [20][21][22][23][24][25]. Proof. Start with Then the expression for G(t) is symmetric in a and b and we can expand G(t) into series in two ways to obtain: On the similar lines, we can show that Comparing the coefficients of t n n! on the right hand sides of the last two equations, we arrive at the desired result (5.1).
Remark 5.1. On setting b = 1 in Theorem 5.1, we get the following corollary.
Corollary 5.1. For n, k ∈ Z and n ≥ 0, the following identity holds true: Proof. Let m,q (ay)b m a n−m t n n! .
In similar method, we can be written as 14 A note on q-analogue of Hermite-poly-Bernoulli numbers . . .
By comparing the coefficients of t n n! on the right hand sides of the last two equations, we arrive at the desired result (5.2). Next making change of index and by equating the coefficients of t to zero in (5.5) and (5.6), we get the result (5.4).
Remark 5.2. By setting y = 0 in Theorem 5.3, we get the following corollary.