Daeyeoul Kim

New approach to the complete sum of products of the twisted (h, q)-Bernoulli numbers and polynomials

Abstract In this paper, by using q-Volkenborn integral[10], the first author[25] constructed new generating functions of the new twisted (h, q)-Bernoulli polynomials and numbers. We define higher-order twisted (h, q)-Bernoulli polynomials and numbers. Using these numbers and polynomials, we obtain new approach to the complete sums of products of twisted (h, q)-Bernoulli polynomials and numbers. p-adic q-Volkenborn integral is used to evaluate summations of the following form: where is the twisted (h, q)-Bernoulli polynomials. We also define new identities involving (h, q)-Bernoulli polnomials and numbers.

ON THE ANALOGS OF BERNOULLI AND EULER NUMBERS, RELATED IDENTITIES AND ZETA AND L-FUNCTIONS

In this paper, by using q-deformed bosonic p-adic integral, we give -Bernoulli numbers and polynomials, we prove Witts type formula of -Bernoulli polynomials and Gauss multiplicative formula for -Bernoulli polynomials. By using derivative operator to the generating functions of -Bernoulli polynomials and generalized -Bernoulli numbers, we give Hurwitz type -zeta functions and Dirichlets type -L-functions; which are interpolated -Bernoulli polynomials and generalized -Bernoulli numbers, respectively. We give generating function of -Bernoulli numbers with order r. By using Mellin transforms to their function, we prove relations between multiply zeta function and -Bernoulli polynomials and ordinary Bernoulli numbers of order r and -Bernoulli numbers, respectively. We also study on -Bernoulli numbers and polynomials in the space of locally constant. Moreover, we define -partial zeta function and interpolation function.

q-Genocchi Numbers and Polynomials Associated with q-Genocchi-Type l-Functions

The main purpose of this paper is to study on generating functions of the Open image in new window -Genocchi numbers and polynomials. We prove new relation for the generalized Open image in new window -Genocchi numbers which is related to the Open image in new window -Genocchi numbers and Open image in new window -Bernoulli numbers. By applying Mellin transformation and derivative operator to the generating functions, we define Open image in new window -Genocchi zeta and Open image in new window -functions, which are interpolated Open image in new window -Genocchi numbers and polynomials at negative integers. We also give some applications of generalized Open image in new window -Genocchi numbers.The main purpose of this paper is to study on generating functions of the -Genocchi numbers and polynomials. We prove new relation for the generalized -Genocchi numbers which is related to the -Genocchi numbers and -Bernoulli numbers. By applying Mellin transformation and derivative operator to the generating functions, we define -Genocchi zeta and -functions, which are interpolated -Genocchi numbers and polynomials at negative integers. We also give some applications of generalized -Genocchi numbers.

CONVOLUTION SUMS ARISING FROM DIVISOR FUNCTIONS

Let s(N) denote the sum of the sth powers of the positive divisors of a positive integer N and let e s(N) = P djN ( 1) d 1 d s with d, N, and s positive integers. Hahn (12) proved that

New identities involving Bernoulli, Euler and Genocchi numbers

Using p-adic integral, many new convolution identities involving Bernoulli, Euler and Genocchi numbers are given.MSC:11B68, 11S80.

A REMARK OF EISENSTEIN SERIES AND THETA SERIES

As a by-product of [5], we produce algebraic integers of certain values of quotients of Eisenstein series. And we consider the relation of and . That is,we show that

Bernoulli numbers, convolution sums and congruences of coefficients for certain generating functions

\Theta_3(0,\tau^n)=\Theta_3(0,\tau),\bigtriangleup(0,\tau)=\bigtriangleup(0,\tau^n)

CONGRUENCES OF THE WEIERSTRASS

and for some .

{\wp}(x)

In this paper, we study the convolution sums involving restricted divisor functions, their generalizations, their relations to Bernoulli numbers, and some interesting applications.MSC: 11B68, 11A25, 11A67, 11Y70, 33E99.

AND

In this paper, we find the coefficients for the Weierstrass and (, , )-functions in terms of the arithmetic identities appearing in divisor functions which are proved by Ramanujan ([23]). Finally, we reprove congruences for the functions and in Hahns article [11, Theorems 6.1 and 6.2].