Some identities of Korobov-type polynomials associated with p-adic integrals on Z p $\mathbb{Z}_{p}$

Dae San Kim,Taekyun Kim

doi:10.1186/s13662-015-0602-8

Abstract

In this paper, we consider Korobov-type polynomials derived from the bosonic and fermionic p-adic integrals on $\mathbb{Z}_{p}$ , and we give some interesting and new identities of those polynomials and of their mixed-types.

Highlights

Let p be a fixed odd prime number
Note that limλ→ Kn(x | λ) = bn(x), where bn(x) are the Bernoulli polynomials of the second kind defined by the generating function t log( + t)
In this paper we introduce two Korobov-type polynomials obtained from the same function, namely the one by performing bosonic p-adic integrals on Zp and the other by carrying out fermionic p-adic integrals on Zp

Summary

Introduction

Let p be a fixed odd prime number. Throughout this paper, Zp, Qp and Cp will denote the ring of p-adic integers, the field of p-adic rational numbers and the completion of the algebraic closure of Qp. The Daehee polynomials of order r are defined by the generating function log( + t) t r ( + t)x =. We introduce the Changhee polynomials of order r given by the generating function t+. Note that limλ→ Kn(x | λ) = bn(x), where bn(x) are the Bernoulli polynomials of the second kind defined by the generating function t log( + t) tn bn(x) n!. We define the higher-order Korobov polynomials given by the generating function λt (t + )λ – Kn(r). For r ∈ N, the λ-Changhee polynomials of order r are defined by the generating function. The Stirling numbers of the second kind are defined by the generating function et – n = n!. The Korobov polynomials (of the first kind) were introduced in [ ] as the degenerate version of the Bernoulli polynomials of the second kind. We will obtain some connections between these new polynomials and Bernoulli polynomials, Euler polynomials, Daehee numbers and Bernoulli numbers of the second kind

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Zp x

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