Generating Functions for q-Apostol Type Frobenius–Euler Numbers and Polynomials

The aim of this paper is to construct generating functions, related to nonnegative real parameters, for q-Eulerian type polynomials and numbers (or q-Apostol type Frobenius–Euler polynomials and numbers). We derive some identities for these polynomials and numbers based on the generating functions and functional equations. We also give multiplication formula for the generalized Apostol type Frobenius–Euler polynomials.


Introduction
Throughout this paper we assume that q ∈ C, the set of complex numbers, with |q| < 1 and We use the following standard notions: and also, as usual R denotes the set of real number, R + denotes the set of positive real number and C denotes the set of complex numbers.
In this section, we define q-Apostol type Frobenius-Euler polynomials and numbers, related to nonnegative real parameters.numbers).
Definition 1 Let a, b ∈ R + (a = b) and u ∈ C {1}.A q-Apostol type Frobenius-Euler numbers H n (u; a, b; λ; q) (λ, q ∈ C) are defined by means of the following generating function: are defined by means of the following generating function: with, of course H n (0; u; a, b; λ; q) = H n (u; a, b; λ; q) where H n (u; a, b; λ; q) denotes the q-Apostol type Frobenius-Euler numbers.
By using the following well-known identity in Equation (2), we derive the following functional equation: By using Equation (3), we arrive at the following theorem: Theorem 1 Let n ∈ N 0 .Then we have By using Theorem 1, one can easily obtain the following result: Corollary 1 Let n ∈ N 0 .Then we have with the usual convention of replacing H j by H j .

Identities
In this section, we derive some identities related to the q-Apostol type Frobenius-Euler numbers and polynomials, using generating functions.
We are now ready to give explicit formulas for computing the q-Apostol type Frobenius-Euler numbers and polynomials.
Theorem 2 Let n ∈ N 0 .Then we have Proof 1 By using Equation (1), we get From the above equation, we obtain Therefore, equating the coefficients of t n n! on both sides of the above equation, we obtain the desired result.
Remark 5.If we put a = 1, u = −1, λ = 1 and b = e, then Theorem 2 reduces to Theorem 1 in [21].If we substitute a = 1, λ = 1 and b = e into Theorem 2, then we obtain an explicit formula, for the q-Frobenius-Euler numbers H n (u; q) = H n (u; 1, e; 1; q), which is given by the following corollary: We give an explicit formula for the q-Apostol type Frobenius-Euler polynomials as follows: Proof 2 Proof of this theorem is the same as that of Theorem 2, so we omit it.

Multiplication Formula
Here we prove multiplication formula for the q-Apostol type Frobenius-Euler numbers and polynomials.This formula is very important to investigate fundamental properties of these polynomials.
Proof 3 Substituting n = l + mf with m = 0, 1, . . ., ∞, l = 1, 2, . . ., f into Equation (2), we get From the above equation, we obtain By using Equation ( 6) in the right side of the above equation, we have Thus, by using the Cauchy product in the above equation and then equating the coefficients of t n n! on both sides of the resulting equation, we get the desired result.
If we put a = 1 in Equation (2), we simplify Theorem 4 as follows: Equating the coefficients of t n n! on both sides of the resulting equation, we obtain: Replacing x by f x in the above equation, we get the following result: Corollary 4 Let n ∈ N 0 .Then we have Remark 7. If we put u = −1, λ = 1 and b = e, and q → 1 in Equation ( 7), Equation (4,13) in [1].