Some identities of Barnes-type special polynomials

Dongkyu Lim,Younghae Do

doi:10.1186/s13662-015-0385-y

Abstract

In this paper, we consider Barnes-type special polynomials and give some identities of their polynomials which are derived from the bosonic p-adic integral or the fermionic p-adic integral on $\mathbb{Z}_{p}$ .

Highlights

As is known, the Bernoulli polynomials of order r are defined by the generating function to be t et – r ext = ∞ B(nr) (x) tn n!. ( ) n=When x =, B(nr) = B(nr)( ) are called the Bernoulli numbers of order r
The purpose of this paper is to investigate several special polynomials related to Barnestype polynomials and give some identities including Witt’s formula of their polynomials
For r ∈ N, the generating function of higher-order Euler polynomials can be derived from the multivariate p-adic fermionic integral on Zp as follows:

Summary

Introduction

The Bernoulli polynomials of order r are defined by the generating function to be t et – r ext =. When x = , B(nr) = B(nr)( ) are called the Bernoulli numbers of order r. Ar = ∈ Cp, the Barnes-Bernoulli polynomials are defined by the generating function to be r t eait – ext =. Ar) are called Barnes Bernoulli numbers (see [ – ]). Bn (x|a , a , From ( ), we obtain the following Witt’s formula for the Barnes-Bernoulli polynomials. Ar by ( ), we obtain the following distribution relation for a Barnes-type Bernoulli polynomial. For r ∈ N, the generating function of higher-order Euler polynomials can be derived from the multivariate p-adic fermionic integral on Zp as follows:

Zp tm m!

Zp n