Hermite based poly-Bernoulli polynomials with a q-parameter

Ugur Duran, Mehmet Acikgoz and Serkan Araci Department of Mathematics, Faculty of Arts and Science, University of Gaziantep, TR-27310 Gaziantep, Turkey E-Mail: mtdrnugur@gmail.com Department of Mathematics, Faculty of Arts and Science, University of Gaziantep, TR-27310 Gaziantep, Turkey E-Mail: acikgoz@gantep.edu.tr Department of Economics, Faculty of Economics, Administrative and Social Science, Hasan Kalyoncu University, TR-27410 Gaziantep, Turkey E-Mail: mtsrkn@hotmail.com Corresponding Author


Introduction
Special polynomials and numbers possess a lot of importances in many …elds of mathematics, physics, engineering and other related disciplines including the topics such as di¤erential equations, mathematical analysis, functional analysis, mathematical physics, quantum mechanics and so on.One of the most considerable polynomials in the theory of special polynomials is the Hermite-Kampé de Fériet (or Gould-Hopper) polynomials (see [1]) and one other is Bernoulli polynomials (see [10], [16]).Nowadays, these type polynomials and their several generalizations have been studied and used by many mathematicians and physicsics, see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] and references therein.Araci et al. [2] introduced a new concept of the Apostol Hermite-Genocchi polynomials by using the modi…ed Milne-Thomson's polynomials and also derived several implicit summation formulae and general symmetric identities arising from di¤erent analytical means and generating functions method.Bretti et al. [4] de…ned multidimensional extensions of the Bernoulli and Appell polynomials by using the Hermite-Kampé de Fériet polynomials and gave the di¤erential equations, satis…ng by the corresponding 2D polynomials, derived from exploiting the factorization method.Bayad et al. [3] considered poly-Bernoulli polynomials and numbers and proved a collection of extremely important and fundamental identities satis…ed by the poly-Bernoulli polynomials and numbers.Cenkci et al. [5] considered poly-Bernoulli numbers and polynomials with a q parameter and developed some aritmetical and number theoretical properties.Dattoli et al. [6] applied the method of generating function to introduce new forms of Bernoulli numbers and polynomials, which were exploited to derive further classes of partial sums involving generalized many index many variable polynomials.Khan et al. [7] introduce the Hermite poly-Bernoulli polynomials and numbers of the second kind and examined some of their applications in combinatorics, number theory and other …elds of mathematics.Kurt et al. [8] studied on the Hermite-Kampé de Fériet based second kind Genocchi polynomials and presented some relatieonships of them.Ozarslan [11] introduced an uni…ed family of Hermite-based Apostol-Bernoulli, Euler and Genocchi polynomials and then, acquired some symmetry identities between these polynomials and the generalized sum of integer powers.Ozarslan also gave explicit closed-form formulae for this uni…ed family and proved a …nite series relation between this uni…cation and 3d-Hermite polynomials.Pathan [12] de…ned a new class of generalized Hermite-Bernoulli polynomials and derived several implicit summation formulae and symmetric identities by using di¤erent analytical means appying generating functions.Pathan et al. [13] introduced a new class of generalized polynomials associated with the modi…ed Milne-Thomson's polynomials ( ) n (x; v) of degree n and order and provided some of their properties.
The usual notations C, R, Z, N and N 0 are referred to the set of all complex numbers, the set of all real numbers, the set of all integers, the set of all natural numbers and the set of all nonnegative integers, respectively, in the content of this paper.
An outline of this paper is as follows.Section 2 contains the de…nitions of the Hermite based poly-Bernoulli polynomials H B (k;j) n;q (x; y) with a q parameter and the Hermite based -Stirling polynomials S ( ;j) 2 (n; m; x; y) of the second kind, and then provides some properties and relationships for H B (k;j) n;q (x; y) and S ( ;j) 2 (n; m; x; y).Section 3 examines several correlations including the Hermite-Kampé de Fériet polynomials, the Hermite based poly-Bernoulli polynomials with a q parameter and the Hermite based -Stirling polynomials of the second kind.

Preliminary Results
The exponential generating function for the Hermite-Kampé de Fériet (or Gould-Hopper) family of polynomials is where j 2 N with j 2. In the case j = 1, the corresponding 2D polynomials are simply expressed by the Newton binomial formula.Upon setting j = 2 in (2.1) gives the two variable Hermite polynomials H n (x; y) and the mentioned polynomials have been used to de…ne 2D extensions of some special polynomails, such as Bernoulli and Euler polynomials (see [6]).
Recalling that the polynomials H (j) n (x; y) are the native solution of the generalized heat equation: @ @t F (x; y) = @ j @x j F (x; y) with F (x; 0) = x n and satis…es the following formula , where b c is Gauss'notation, and represents the maximum integer which does not exceed a number in the square brackets. For We always assume jtj < 1 along this paper.When k = 1, Li 1 (t) = log (1 t).In the case k 0, Li k (t) are the rational functions: The Hermite based poly-Bernoulli polynomials with a q parameter.Let n; k 2 Z in conjunction with n 0 and k > 0 and let q 2 R f0g.We introduce the Hermite based poly-Bernoulli polynomials with a q parameter via the following exponential generating function to be Upon setting x = y = 0, we then get H B (k;j) n;q (0; 0) =: H B (k;j) n;q which is called the poly-Bernoulli numbers with a q parameter (see [5]).
Some special cases of H B (k;j) n;q (x; y) are listed by the following consecutive remarks.
Remark 2. In the case q = 1, H B (k;j) n;q (x; y) reduces to the the Hermite based poly-Bernoulli polynomials H B (k;j) n (x; y) (see [11]).
The addition formula for H B (k;j) n;q (x; y) is provided in the following proposition.
An immediate output of Proposition 1 is stated in the following corallary.
Corollary 1.The following holds true: The derivative properties of H B (k;j) n;q (x; y) are stated in the following proposition.
Proof.Using the derivative properties of H B (k;j) n;q (x; y) given in Proposition 2, we easily get the asserted results.So, we omit them.
We have the following proposition.
Proposition 4. The following formula is valid: Proof.By (2.3), we have n jm;q (x) which implies the desired result (2.6).(n; m; x; y) reduces to the S 2 (n; m; x) called the weighted -Stirling numbers of the second kind (see [5]).
We give some relations and properties belonging to the Hermite based -Stirling polynomials of the second kind by the following consecutive propositions.Proposition 5. We have where (n) l equals to n (n 1) (n 2) (n l + 1) called the falling factorial function.

Main Results
This part includes our main results.
A correlation including H B (k;j) n;q (x; y) and H (j) n (x; y) is given by the following theorem.
Theorem 1.The polynomials H B (k;j) n;q (x; y) and H (j) n (x; y) satis…es the following relation: r H (j) n (x + q rq; y) .
A correlation between H B (k;j) n;q (x; y) and S (j) 2 (n; m; x; y) is stated in the following theorem.
From (2.2), we readily derive that An relation for the Hermite based poly-Bernoulli polynomials with a q parameter is given by the following theorem.
n m;q (x; y) q m (x + q) H B (k;j) n;q (x; y) Proof.In the light of (3.5), we get Di¤erentiate both sides of (3.2) with respect to t, we derive Li k 1 1 e qt q e (x+q)t+yt j +qLi k 1 e qt q h (x + q) e (x+q)t+yt j + yjt j 1 e (x+q)t+yt j i = q 2 e qt 1 Li k 1 1 e qt q e (x+q)t+yt j +e qt (x + q) qLi k 1 e qt q e qt 1 e (x+q)t+yt j (x + q) qLi k 1 e qt q e qt 1 e (x+q)t+yt j +e qt yjt j 1 qLi k 1 e qt q e qt 1 e (x+q)t+yt j yjt j 1 qLi k 1 e qt q e qt 1 e (x+q)t+yt j = q 1 X n=0 H B (k 1;j) n;q (x; y) t n n!
H B (k;j) n;q (x; y) t n n! (x + q) 1 X n=0 H B (k;j) n;q (x; y) n m;q (x; y) q m H B (k;j) n;q (x; y) n m;q (x; y) q m H B (k;j) n;q (x; y) !t n+j 1 n! , which gives the desired result (3.6).
Here, we give the following theorem.n; m; x q + 1; y q j q n S q 1 2 (k; m; 1) .
Proof.By inspiring the proof given by Cenkci and Komatsu [5], by (2.3), we write

Remark 6 .
m; x; y) called the Hermite based Stirling polynomials of the second kind.Letting y = 0, S ( ;j) 2

Theorem 4 .
Let n 2 N 0 and k 2 N. We then haveH B ( k;j)
Hermite based -Stirling polynomials of the second kind.We introduce the Hermite based -Stirling polynomials of the second kind is de…ned by