Arithmetic Identities Involving Bernoulli and Euler Numbers

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 [1–8]): The Euler polynomials, , are defined by the generating function as follows (see [1–16]): 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, 17–19]). 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 [1–8, 13–16]. 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 [1–19], 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),