Arithmetic Identities Involving Bernoulli and Euler Numbers

Abstract

The purpose of this paper is to give some arithmatic identities for the Bernoulli and Euler numbers. These identities are derived from the several p -adic integral equations on [subscript] ...p[/subscript] .

(ProQuest: ... denotes non-US-ASCII text omitted.)

H.-M. Kim 1 and D. S. Kim 2

Recommended by A. Bayad

1, Department of Mathematics, Kookmin University, Seoul 136-702, Republic of Korea
2, Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea

Received 12 June 2012; Accepted 23 October 2012

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Let p be a fixed odd prime number. Throughout this paper, _{... p} , _{... p} , and _{... p} will denote the ring of p -adic rational integers, the field of p -adic rational numbers, and the completion of algebraic closure of _{... p} , respectively. The p -adic norm is normalized so that |p _{| p} =1 /p . Let ... be the set of natural numbers and _{... +} = ... ∪ {0 } .

Let UD ( _{... p} ) be the space of uniformly differentiable functions on _{... p} . For f ∈UD ( _{... p} ) , the bosonic p -adic integral on _{... p} is defined by [figure omitted; refer to PDF] and the fermionic p -adic integral on _{... p} is defined by Kim as follows (see [ 1- 8]): [figure omitted; refer to PDF]

The Euler polynomials, _{E n} (x ) , are defined by the generating function as follows (see [ 1- 16]): [figure omitted; refer to PDF] In the special case, x =0 , _{E n} (0 ) = _{E n} is called the n th Euler number.

By ( 1.3) and the definition of Euler numbers, we easily see that [figure omitted; refer to PDF] with the usual convention about replacing ^{E l} by _{E l} (see [ 10]). Thus, by ( 1.3) and ( 1.4), we have [figure omitted; refer to PDF] where _{δ k ,n} is the Kronecker symbol (see [ 9, 10, 17- 19]).

From ( 1.2), we can also derive the following integral equation for the fermionic p -adic integral on _{... p} as follows: [figure omitted; refer to PDF] see [ 1, 2]. By ( 1.3) and ( 1.6), we get [figure omitted; refer to PDF] Thus, by ( 1.7), we have [figure omitted; refer to PDF] see [ 1- 8, 13- 16].

The Bernoulli polynomials, _{B n} (x ) , are defined by the generating function as follows: [figure omitted; refer to PDF] see [ 18]. In the special case, x =0 , _{B n} (0 ) = _{B n} is called the n th Bernoulli number. From ( 1.9) and the definition of Bernoulli numbers, we note that [figure omitted; refer to PDF] see [ 1- 19], with the usual convention about replacing ^{B l} by _{B l} . By ( 1.9) and ( 1.10), we easily see that [figure omitted; refer to PDF] see [ 13].

From ( 1.1), we can derive the following integral equation on _{... p} : [figure omitted; refer to PDF] where _{f 1} (x ) =f (x +1 ) and f ' (0 ) = ( df ( x ) / dx ) _{| x =0} .

By ( 1.12), we have [figure omitted; refer to PDF] Thus, by ( 1.13), we can derive the following Witt's formula for the Bernoulli polynomials: [figure omitted; refer to PDF]

In [ 19], it is known that for k ,m ∈ _{... +} , [figure omitted; refer to PDF] where ( k j ) =0 if j <0 or j >k .

The purpose of this paper is to give some arithmetic identities involving Bernoulli and Euler numbers. To derive our identities, we use the properties of p -adic integral equations on _{... p} .

2. Arithmetic Identities for Bernoulli and Euler Numbers

Let us take the bosonic p -adic integral on _{... p} in ( 1.15) as follows: [figure omitted; refer to PDF] On the other hand, we get [figure omitted; refer to PDF] By ( 2.1) and ( 2.2), we get [figure omitted; refer to PDF]

Therefore, by ( 2.3), we obtain the following theorem.

Theorem 2.1.

For k ,m ∈ _{... +} , one has [figure omitted; refer to PDF]

Now we consider the fermionic p -adic integral on _{... p} in ( 1.15) as follows: [figure omitted; refer to PDF] On the other hand, we get [figure omitted; refer to PDF] By ( 2.5) and ( 2.6), we get [figure omitted; refer to PDF] Therefore, by ( 2.7), we obtain the following theorem.

Theorem 2.2.

For k ,m ∈ _{... +} , one has [figure omitted; refer to PDF]

Replacing x by (1 -x ) in ( 1.15), we have the identity: [figure omitted; refer to PDF] Let us take the bosonic p -adic integral on _{... p} in ( 2.9) as follows: [figure omitted; refer to PDF]

On the other hand, we see that [figure omitted; refer to PDF] By ( 2.10) and ( 2.11), we get [figure omitted; refer to PDF] Therefore, by ( 2.12), we obtain the following theorem.

Theorem 2.3.

For k ,m ∈ _{... +} , one has [figure omitted; refer to PDF]

We consider the fermionic p -adic integral on _{... p} in ( 2.9) as follows: [figure omitted; refer to PDF] On the other hand, we get [figure omitted; refer to PDF] By ( 2.14) and ( 2.15), we obtain the following theorem.

Theorem 2.4.

For k ,m ∈ _{... +} , one has [figure omitted; refer to PDF]

Acknowledgment

This Research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology (2012R1A1A2003786).

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