Generating function for q-Eulerian polynomials and their decomposition and applications

Abstract

The aim of this paper is to define a generating function for q-Eulerian polynomials and numbers attached to any character χ of the finite cyclic group G. We derive many functional equations, q-difference equations and partial deferential equations related to these generating functions. By using these equations, we find many properties of q-Eulerian polynomials and numbers. Using the generating element of the finite cyclic group G and the generating element of the subgroups of G, we show that the generating function with a conductor f can be written as a sum of the generating function with conductors which are less than f . MSC: 05A40; 11B83; 11B68; 11S80

Fourier expansion and integral representation generalized Apostol-type Frobenius–Euler polynomials

“…These polynomials are commonly said to be of Euler type, and they have been studied by various authors in different applications of practical importance (see [1,12,21]). On the other hand, the Hurwitz-Lerch zeta function (z, s, a) is defined as (see [15, p. 296…”

The actions on the generating functions for the family of the generalized Bernoulli polynomials

“…Recently in [1], the authors study the periodic function to decompose the q-Eulerian numbers and polynomials. This decomposition provided us to compute q-Apostol-type Frobenius-Euler polynomials and numbers more easily.…”

“…The classical Bernoulli polynomials B n (x) and the classical Euler polynomials E n (x) are usually defined by means of the following generating functions, respectively (cf. [1]- [23]):…”

“…where B n (λ) denotes the so-called Apostol-Bernoulli Bernoulli polynomials (cf. [1]- [23]). The Frobenius-Euler numbers H n (u) are defined by means of the following generating function:…”

A Numerical Investigation on the Structure of the Zeros of Euler Polynomials

“…Mathematicians have studied different kinds of the Euler, Bernoulli, Tangent, and Genocchi numbers and polynomials. Recently, many authors have studied in the area of the -analogues of these numbers and polynomials (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]). Using computer, a realistic study for the zeros of Euler polynomials ( ) is very interesting.…”