On Some Arithmetical Properties of the Genocchi Numbers and Polynomials

Kyoung Ho Park ,Young‐Hee Kim

doi:10.1155/2008/195049

Abstract

We investigate the properties of the Genocchi functions and the Genocchi polynomials. We obtain the Fourier transform on the Genocchi function. We have the generating function of -Genocchi polynomials. We define the Cangul-Ozden-Simsek's type twisted -Genocchi polynomials and numbers. We also have the generalized twisted -Genocchi numbers attached to the Dirichlet's character . Finally, we define zeta functions related to -Genocchi polynomials and have the generating function of the generalized -Genocchi numbers attached to .

Highlights

After Carlitz introduced an interesting q-analogue of Frobenius-Euler numbers in 1, qBernoulli and q-Euler numbers and polynomials have been studied by several authors
We investigate some arithmetical properties of the Genocchi functions and the Genocchi polynomials
In 8, Cangul-Ozden-Simsek constructed new generating functions of the twisted h, q -extension of twisted Euler polynomials and numbers attached to the Dirichlet

Summary

Introduction

After Carlitz introduced an interesting q-analogue of Frobenius-Euler numbers in 1 , qBernoulli and q-Euler numbers and polynomials have been studied by several authors. In 3 , Kim derived the q-analogs of the Genocchi numbers and polynomials by constructing q-Euler numbers. He gave some interesting relations between q-Euler and q-Genocchi numbers. Kim 7 investigated the properties of the Euler functions and derived the interesting formula related to the infinite series by using the Fourier transform for the Euler function. In 8 , Cangul-Ozden-Simsek constructed new generating functions of the twisted h, q -extension of twisted Euler polynomials and numbers attached to the Dirichlet. We define the Cangul-Ozden-Simsek type twisted h, q -Genocchi polynomials and numbers. We define zeta functions related to h, q -Genocchi polynomials and we have the generating function of the generalized h, q -Genocchi numbers attached to χ. We say that f is uniformly differential function at a point a ∈ Zp, and we write f ∈ UD Zp , if the difference quotients, Ff x, y f x −f y /x −y have a limit f a as x, y → a, a

Iq f f x dμq x

Gn n!