Multifarious results for q-Hermite based Frobenius type Eulerian polynomials

In this paper, a new class of q-Hermite based Frobenius type Eulerian polynomials is introduced by means of generating function and series representation. Several fundamental formulas and recurrence relations for these polynomials are derived via di¤erent generating methods. Furthermore, diverse correlations including the q-Apostol-Bernoulli polynomials, the q-Apostol-Euler poynoomials, the q-Apostol-Genocchi polynomials and the q-Stirling numbers of the second kind are also established by means of the their generating functions.


Introduction
The subject of q-calculus started appearing in the nineteenth century due to its applications in various fields of mathematics, physics and engineering. One of the most popular studies in q-calculus is the q-extension of the some special polynomials such as Bernoulli, Eulerian, Genocchi, Euler polynomials (cf.  and the references cited therein). The classical Eulerian polynomials are firstly considered by Leonhard Euler in his Remarques sur un beau rapport entre les séries des puissances tant directes que réciproques in 1749 (first printed in 1765) in which he defined a method of computing values of the zeta function at negative integers. Then, the foregoing polynomials have been studied and investigated extensively until now, cf. [3,4,12,13]. Moreover, the Frobenius-type Eulerian polynomials are introduced and studied in [22]. Kim [9] studied new q-extensions of Euler numbers and polynomials and investigated some properties of symmetries of these q-Euler polynomials by using q-derivatives and q-integrals. Kim [10] introduced a Daehee constant, the so-called q-extension of the Napier constant, and considered the Daehee formula associated with the q-extensions of trigonometric functions, and then derived the q-extensions of sine and cosine functions from this Daehee formula and the q-calculus related to the q-extensions of sine and cosine functions. In this paper, we perform to give a novel class of q-generalization of the Hermite-based Frobenius-type Eulerian polynomials and to derive multifarious correlation, implicit summation formula, identities, explicit formulas and recurrence relations for the mentioned polynomials by means of the series manipulation methods. Furthermore, we investigate some correlations covering the q-Apostol-Bernoulli polynomials, q-Apostol-Euler polynomials, q-Apostol-Genocchi polynomials and q-Stirling numbers of the second kind for the q-Hermite-based Frobenius-type Eulerian polynomials. The definitions and notations of q-calculus reviewed here are taken from the references [5-10, 14-17, 20]: The q-analogue of the shifted factorial (a) n is given by The q-analogues of a complex number a and of the factorial function are given by The q-analogue of the function (x + y) n q is given by The q-analogues of exponential function are given by Moreover, the functions e q (x) and E q (x) satisfy the following properties: where the q-derivative D q f (x) of a function f at a point 0 = x ∈ C is defined as follows: For any two arbitrary functions f (x) and g(x), the q-derivative operator D q satisfies the following product and quotient relations: . (1.6) The Apostol-type q-Bernoulli polynomials B (α) n,q (x, y; λ) of order α, the Apostol-type q-Euler polynomials E (α) n,q (x, y; λ) of order α and the Apostol-type q-Genocchi polynomials G (α) n,q (x, y; λ) of order α are defined by means of the following generating function (see [7,8,[14][15][16][17]): n,q (x, y; λ) t n n! , (|t + log λ|) < 2π, 1 α = 1, n,q (x, y; λ) t n n! , (|t + log λ|) < π, 1 α = 1 (1.8) and 2t n,q (x, y; λ) t n n! , |t + log λ|) < π, 1 α = 1. (1.9) Clearly, we have The Frobenius-type Eulerian polynomials A (α) n (x; λ) of order α ∈ C are defined by means of the following generating function as follows (see [22]): where λ is a complex number with λ = 1. The number A n is given by are called the Frobenius-type Eulerian numbers (see [21,22]). Clearly, we have In recent days, a new type of q-Hermite polynomials are considered in [16,20], which is a particular member of the q-Appell family [1]. The q-Appell polynomials are defined by means of the following generating function: n,q (0) are the continuous q-Hermite numbers defined by 2 q-Hermite-based Frobenius-type Eulerian polynomials In this section, we define q-Hermite-based Frobenius-type Eulerian polynomials (qHbF tEp) n,q (x, y; λ) by means of the generating function and series representation. Certain relations for these polynomials are also derived by using various identities. We now ready to start in conjunction with the following definition.
n,q (x, y; λ) of order α are defined by means of the following generating function: n (λ) are called the n-th Frobenius-type Eulerian numbers of order α.
Proof. Consider the generating function (2.1), we have n,q (x, y; λ) are called the q-Hermite based Frobenius polynomials, which is defined by Riyasat and Khan [20] and comparing the coefficients of t n , we arrive at the required result (2.3).  n,q (x, y; ; λ) of order α holds true: (2.10) Proof. We set .
From the above equation, we see that , which on using equations (1.10) and (2.1) in both sides, we have Applying the Cauchy product rule in the above equation and then equating the coefficients of like powers of t in both sides of the resultant equation, assertion (2.10) follows.
Theorem 2.6. The following relation for the q-Hermite-based Frobenius-type Eulerian polynomials H A (α,s) n,q (x, y; λ) of order α holds true: Proof. Consider the following identity Evaluating the following fraction using above identity, we find Applying the Cauchy product rule in the above equation and then equating the coefficients of like powers of t in both sides of the resultant equation, assertion (2.11) follows.
Theorem 2.7. The following relation for the q-Hermite-based Frobenius-type Eulerian polynomials H A (α,s) n,q (x, y; λ) of order α holds true: Applying the Cauchy product rule in the above equation and then equating the coefficients of like powers of t in both sides of the resultant equation, assertion (2.12) follows.
Proof. Taking α = 1 and then applying q-derivative on both sides of generating function (2.1), it follows that which on performing differentiation in left-hand side, using formula (1.6), yields which on making use of the Cauchy product rule in the right-hand side and comparing the coefficients of Proof. We replace t by t + w and rewrite the generating function (2.1) as , (see [18,19]). (3.2) 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 [23, p.52(2)] f (n + m) x n n! y m m! , (3.5) in the left-hand side becomes ∞ n,m=0 Now replacing k by k − n, and l by l − m in the left-hand side of (3.6), we get ∞ k,l=0 k,l n,m=0 Finally on equating the coefficients of the like powers of t and w in the above equation, we get the required result.
Finally, equating the coefficients of the like powers of t n , we get (3.12).
Furthermore, we have given several relationships covering the q-Apostol-Bernoulli polynomials, q-Apostol-Euler polynomials, q-Apostol-Genocchi polynomials and q-Stirling numbers of the second kind associated with the q-Hermite-based Frobenius-type Eulerian polynomials.