%0 Journal Article
%T Arithmetic Identities Involving Bernoulli and Euler Numbers
%A H.-M. Kim
%A D. S. Kim
%J International Journal of Mathematics and Mathematical Sciences
%D 2012
%I Hindawi Publishing Corporation
%R 10.1155/2012/689797
%X The purpose of this paper is to give some arithmatic identities for the Bernoulli and Euler numbers. These identities are derived from the several -adic integral equations on . 1. Introduction Let be a fixed odd prime number. Throughout this paper, , , and will denote the ring of -adic rational integers, the field of -adic rational numbers, and the completion of algebraic closure of , respectively. The -adic norm is normalized so that . Let be the set of natural numbers and . Let be the space of uniformly differentiable functions on . For , the bosonic -adic integral on is defined by and the fermionic -adic integral on is defined by Kim as follows (see [1每8]): The Euler polynomials, , are defined by the generating function as follows (see [1每16]): In the special case, , is called the th Euler number. By (1.3) and the definition of Euler numbers, we easily see that with the usual convention about replacing by (see [10]). Thus, by (1.3) and (1.4), we have where is the Kronecker symbol (see [9, 10, 17每19]). From (1.2), we can also derive the following integral equation for the fermionic -adic integral on as follows: see [1, 2]. By (1.3) and (1.6), we get Thus, by (1.7), we have see [1每8, 13每16]. The Bernoulli polynomials, , are defined by the generating function as follows: see [18]. In the special case, , is called the th Bernoulli number. From (1.9) and the definition of Bernoulli numbers, we note that see [1每19], with the usual convention about replacing by . By (1.9) and (1.10), we easily see that see [13]. From (1.1), we can derive the following integral equation on : where and . By (1.12), we have Thus, by (1.13), we can derive the following Witt＊s formula for the Bernoulli polynomials: In [19], it is known that for , where if or . 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 -adic integral equations on . 2. Arithmetic Identities for Bernoulli and Euler Numbers Let us take the bosonic -adic integral on in (1.15) as follows: On the other hand, we get By (2.1) and (2.2), we get Therefore, by (2.3), we obtain the following theorem. Theorem 2.1. For , one has Now we consider the fermionic -adic integral on in (1.15) as follows: On the other hand, we get By (2.5) and (2.6), we get Therefore, by (2.7), we obtain the following theorem. Theorem 2.2. For , one has Replacing by in (1.15), we have the identity: Let us take the bosonic -adic integral on in (2.9) as follows: On the other hand, we see that By (2.10) and (2.11), we get Therefore, by (2.12),
%U http://www.hindawi.com/journals/ijmms/2012/689797/