Symmetry Properties of Higher-Order Bernoulli Polynomials

Abstract

We investigate properties of identities and some interesting identities of symmetry for the Bernoulli polynomials of higher order using the multivariate -adic invariant integral on .

1. Introduction

Let be a fixed prime number. Throughout this paper and will, respectively, denote the ring of -adic rational integers, the field of -adic rational numbers, and the completion of algebraic closure of . For , we use the notation . Let be the space of uniformly differentiable functions on and let be the normalized exponential valuation of with . For with , the -Volkenborn integral on is defined as

(1.1)

(see [1, 2]). The ordinary -adic invariant integral on is given by

(1.2)

(see [1–15]). Let . Then we easily see that

(1.3)

From (1.3), we can derive

(1.4)

(see [2, 8–10]), where are the th Bernoulli numbers.

By (1.2) and (1.3), we easily see that

(1.5)

where for

It is known that the Bernoulli polynomials are defined by

(1.6)

where are called the th Bernoulli polynomials. The Bernoulli polynomials of order , denoted , are defined as

(1.7)

(see [3–6]). Then the values of at are called the Bernoulli numbers of order . When , the polynomials or numbers are called the Bernoulli polynomials or numbers. The purpose of this paper is to investigate some interesting properties of symmetry for the multivariate -adic invariant integral on . From the properties of symmetry for the multivariate -adic invariant integral on , we derive some interesting identities of symmetry for the Bernoulli polynomials of higher order.

2. Symmetry Properties of Higher-Order Bernoulli Polynomials

Let . Then we define

(2.1)

From (2.1), we note that

(2.2)

where

In (2.1), we note that is symmetric in . By (2.1), we see that

(2.3)

It is easy to see that

(2.4)

From (2.1), (2.3), and the above formula, we can derive

(2.5)

By the symmetry of in and , we see that

(2.6)

By comparing the coefficients on both sides of (2.5) and (2.6), we obtain the following theorem.

Theorem 2.1.

For , one has

(2.7)

Let and in (2.7). Then we have the following corollary.

Corollary 2.2.

For , one has

(2.8)

If we take in (2.8), then we also obtain the following corollary.

Corollary 2.3.

For one has

(2.9)

By the definition of , we easily see that

(2.10)

From the symmetric property of in , we note that

(2.11)

By comparing the coefficients on both sides of (2.10) and (2.11), we obtain the following theorem.

Theorem 2.4.

For , one has

(2.12)

Let and in (2.12). Then we obtain the following Corollary 2.5.

Corollary 2.5.

For , one has

(2.13)

From (2.12), we can get the well-known result due to Raabe:

(2.14)