Some properties of q-Hermite Fubini numbers and polynomials

Abstract. The main purpose of this paper is to introduce a new class of q-HermiteFubini numbers and polynomials by combining the q-Hermite polynomials and qFubini polynomials. By using generating functions for these numbers and polynomials, we derive some alternative summation formulas including powers of consecutive q-integers. Also, we establish some relationships for q-Hermite-Fubini polynomials associated with q-Bernoulli polynomials, q-Euler polynomials and q-Genocchi polynomials and q-Stirling numbers of the second kind.


Introduction
The subject of q-calculus started appearing in the nineteenth century due to its applications in various fields of mathematics, physics and engineering.The definitions and notations of q-calculus reviewed here are taken from (see [1]): The q-analogue of the shifted factorial (a) n is given by (a; q) 0 = 1, (a; q) n = n−1 ∏ m=0 (1 − q m a), n ∈ N.
The Gauss q-binomial coefficient is given by The q-analogue of the function (x + y) n q is given by The q-analogue of exponential function are given by q n(n−1)/2 x n [n] q != (−(1 − q)x; q) ∞ , 0 <| q < 1; x ∈ C.
(1.3)Moreover, the functions e q (x) and E q (x) satisfy the following properties: D q e q (x) = e q (x), D q E q (x) = E q (qx), (1.4) 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 q-derivative operator D q satisfies the following product and quotient relations: The q-Hermite polynomials are special or limiting case of the orthogonal polynomials as they contain no parameter other than q and appears to be at the bottom of a hierarchy of the classical polynomials [2].The q-Hermite polynomials constitute a 1-parameter family of orthogonal polynomials, which for q = 1 reduce to the well known Hermite polynomials.We recall that the q-Hermite polynomials H n,q (x) is defined by means of the following generating function (see [9]): The q-Bernoulli polynomials B (α) n,q (x, y) of order α, the q-Euler polynomials E (α) n,q (x, y) of order α and the q-Genocchi polynomials G (α) n,q (x, y) of order α are defined by means of the following generating function (see [1][2][8][9][10][11]): ( 2 e q (t) + 1 ( 2t e q (t) + 1 (1.9) Clearly, we have n,q .Geometric polynomials (also known as Fubini polynomials) are defined as follows (see [3]): } is the Stirling number of the second kind (see [5]).
For x = 1 in (1.10), we get n th Fubini number (ordered Bell number or geometric number) F n [4,6,7,13] is defined by The exponential generating functions of geometric polynomials is given by (see [3]): and related to the geometric series (see [3]): Let us give a short list of these polynomials and numbers as follows: and Geometric and exponential polynomials are connected by the relation (see [3]): ( The manuscript of this paper as follows: In section 2, we consider generating functions for q-Hermite-Fubini numbers and polynomials and give some properties of these numbers and polynomials.In section 3, we derive summation formulas of q-Hermite-Fubini numbers and polynomials and some relationships between q-Bernoulli polynomials, q-Euler polynomials and q-Genocchi polynomials and Stirling numbers of the second kind.

A q-analogue type of Hermite-Fubini numbers and polynomials
In this section, we define q-analogue type of Hermite-Fubini polynomials and obtain some basic properties which gives us new formula for H F n,q (x; y).Moreover, we shall consider the sum of products of two q-analogue type of Hermite-Fubini polynomials.The sum of products of various polynomials and numbers with or without binomial coefficients have been studied by (see [4,6,7,13]): We introduce q-Hermite-based Fubini polynomials in two variables by means of the following generating function: Taking x = 0, y = 1 in (2.1), we get , where H F n,q are the q-Hermite-based Fubini numbers.
When investigating the connection between q-Hermite polynomials H n,q (x) and q-Fubini polynomials F n,q (y) of importance is the following theorem.Theorem 2.1.The following formula for q-Hermite-based Fubini polynomials holds true: 2) Proof.Using definition (2.1), we have Comparing the coefficients of t n [n]q! yields (2.2).Utilizing equation (1.6) in the l.h.s. of generating function (2.1), it follows that which on applying the Cauchy product rule in the l.h.s. and then comparing the coefficients of same powers of t in both sides of resultant equation yield assertion (2.3).
Proposition 2.1.The following formula for q-Hermite-based Fubini polynomials holds true: D q,t e q (xt) = xe q (xt) Theorem 2.2.For n ≥ 0, the following formula for q-Hermite-based Fubini polynomials holds true: Proof.We begin with the definition (2.1) and write F q (t)e q (xt).
Then using the definition of q-Hermite polynomials H n,q (x) and (2.1), we have Finally, comparing the coefficients of t n n! , we get (2.5).Theorem 2.3.For n ≥ 0, the following formula for q-Hermite-based Fubini polynomials holds true:

Preprints
Proof.The products of (2.1) can be written as By equating the coefficients of t n [n]q! on both sides, we get (2.6).Theorem 2.4.For n ≥ 0, the following formula for q-Hermite-based Fubini polynomials holds true: (2.7) Proof.From (2.1), we have Comparing the coefficients of t n [n]q! on both sides, we obtain (2.7).Remark 2.1.On setting x = 0 and x = −1 in Theorem 2.4, we find and (2.9)

Main results
In this section,, we prove the following result involving q-Hermite-Fubini polynomials H F n,q (x; y) by using series rearrangement techniques and considered its special case.Also we obtain some relationships for q-Hermite Fubini polynomials related to q-Bernoulli polynomials, q-Euler polynomials and q-Genocchi polynomials and Stirling numbers of the second kind in Theorem 3.1.The following formula for q-Hermite-based Fubini polynomials holds true: Proof.Replacing t by t + u in (2.1) and then using the formula [12,p.52(2)]: in the resultant equation, we find the following generating function for the Hermite-Fubini polynomials H F n (x, y; z): Replacing x by w in the above equation and equating the resultant equation to the above equation, we find On expanding exponential function (3.4) gives which on using formula (3.2) in the first summation on the left hand side becomes (3.6) Now replacing q by q − n, l by l − p and using the lemma ([12, p.100(1)]): in the l.h.s. of (3.6), we find The following summation formula for Hermite-Fubini polynomials H F n (x, y; z) holds true: Remark 3.2.Replacing w by w + x in (3.9), we obtain Theorem 3.2.The following formula for q-Hermite-based Fubini polynomials holds true: Proof.Using the generating function (2.1), we have ) (e q (t) − 1)F q (t)e q (xt) Finally, equating the coefficients of the like powers of t on both sides, we get (3.11).
Theorem 3.3.Each of the following relationships holds true: where B n,q (x) is q-Bernoulli polynomials.
By using Cauchy product and comparing the coefficients of t n [n]q! , we arrive at the required result (3.12).Theorem 3.4.Each of the following relationships holds true: where E n,q (x) is the q-Euler polynomials.