An invariant p-adic q-integral associated with q-Euler numbers and polynomials

Abstract

The purpose of this paper is to consider q-Euler numbers and polynomials which are q-extensions of ordinary Euler numbers and polynomials by the computations of the p-adic q-integrals due to T. Kim, cf. [1,3,6,12], and to derive the "complete sums for q-Euler polynomials" which are evaluated by using multivariate p-adic q-integrals. These sums help us to study the relationships between p-adic q-integrals and nonarchimedean combinatorial analysis.

q-Hardy–Berndt type sums associated with q-Genocchi type zeta and q-l-functions

“…[7,8,18,21,16,24,23,37,36,39,40,42]. A function l G (s, χ ) is the so-called ordinary Genocchi-type l-function.…”

q-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients

“…For x ∈ C p , we use the notation [x] q = 1−q x 1−q , cf. [1][2][3][4][5][6]. We say that f is a uniformly differentiable function at a point a ∈ Z p and denote…”

Euler Numbers and Polynomials Associated with Zeta Functions

“…where f ∈ U D(Z p ) = the space of uniformly differentiable function on Z p with values in C p , cf. [1,13,27,28,29,30,31]. In view of notation, I −1 can be written symbolically as…”

“…For s ∈ C, Euler zeta function and Hurwitz's type Euler zeta function are defined by [1,11,12,20,21,22]. Thus, we note that Euler zeta functions are entire functions in whole complex s-plane and these zeta functions have the values of Euler numbers or Euler polynomials at negative integers.…”

“…Thus, we note that Euler zeta functions are entire functions in whole complex s-plane and these zeta functions have the values of Euler numbers or Euler polynomials at negative integers. That is, [1,11,12,20,21,22] .…”