On the Higher‐Order q‐Euler Numbers and Polynomials with Weight α

Abstract

The main purpose of this paper is to present a systemic study of some families of higher-orderq-Euler numbers and polynomials with weightα. In particular, by using the fermionicp-adicq-integral onℤp, we give a new concept ofq-Euler numbers and polynomials with weightα.

Identities involving Frobenius–Euler polynomials arising from non-linear differential equations

“…n (u) are called the n-th Frobenius-Euler numbers of order r (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14]16]). In this paper we derive non-linear differential equations from (1) and we study the solutions of non-linear differential equations.…”

Some Identities Involving Euler Polynomials Arising from a Non-linear Differential Equation

“…where B m (x) is the well-known Bernoulli polynomials. (See [2,4,5,12]). For r ∈ N, the Euler polynomials E (r)…”

“…n are called the n-th Euler numbers of order r. (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]17])…”

ON THE EXTENDED q-EULER NUMBERS AND POLYNOMIALS OF HIGHER-ORDER WITH WEIGHT

“…with the usual convention about replacing Ẽ (α) q n byẼ (α) n,q (see [5]). The q-Euler polynomials with weight α also defined bỹ…”

“…Then the fermionic p-adic q-integral on Z p is defined by Kim as follows ( [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]):…”