Euler Numbers and Polynomials Associated with Zeta Functions

Abstract

We consider the q-analogue of Euler zeta function which is defined byIn this paper, we give the q-extension of Euler numbers which can be viewed as interpolating of the above q-analogue of Euler zeta function at negative integers , in the same way that Riemann zeta function interpolates Bernoulli numbers at negative integers. Also, we will treat some identities of the q-extension of the Euler numbers by using fermionic p-adic q-integration on Z p .

CERTAIN RESULTS ON THE q-GENOCCHI NUMBERS AND POLYNOMIALS

“…By using definition of the geometric series in (3.1), we easily see that which is well-known Euler-Zeta function (see [7]). By (3.4) and (3.5), we get ζ (−n, x : q) = G n+1,q (x) n + 1 .…”

“…The integral of η on Z p will be defined as the limit (n → ∞) of these sums, when it exists. The p-adic q-integral of function η ∈ U D (Z p ) is defined by T. Kim in [7], [11], [16] by…”

Dynamics of the zeros of analytic continued polynomials and differential equations associated with q-tangent polynomials

“…Using software algorithm, researchers can explore theoretical concepts and numerical experiments much more easily than in the past. There have been lots of research by mathematician in different kinds of the Tangent, Euler, Bernoulli, and Genocchi numbers and polynomials (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]). Numerical developments as well as experiments of Bernoulli polynomials, Euler polynomials, Genocchi polynomials, and Tangent polynomials have been studied with the significant progress in mathematics and computer science as one of the interesting subjects for the development of computer algorithms.…”

Fourier series of sums of products of Genocchi functions and their applications

“…For elementary facts about Fourier analysis, the reader may refer to any book (for example, see [8,17,18,22]). As to γ m (< x >), we note that the polynomial identity (1.2) follows immediately from Theorems 4.1 and 4.2, which is in turn derived from the Fourier series expansion of γ m (< x >),…”