Identities on the Weighted -Bernoulli Numbers of Higher Order

We give a new construction of the weighted q-Bernoulli numbers and polynomials of higher order by using multivariate -adic -integral on . 1. Introduction Let be a fixed prime number. Throughout this paper, , and will, respectively, denote the ring of rational integers, the ring of -adic rational integers, the field of -adic rational numbers, the complex number field, and the completion of the algebraic closure of . The -adic norm of is defined by where with . In this paper, we assume and . Let with and let . Note that (see [1–13]). Recently, the -Bernoulli numbers with weight are defined by with the usual convention about replacing by , (see [4]). The -Bernoulli polynomials with weight are also defined by as For the space of uniformly differentiable functions on , the -adic -integral on is defined by (see [4–12]). From (1.3), we note that where , (see [1–12]). We have the Witt formula for the -Bernoulli numbers and polynomials with weight as follows (see [4, 5, 12]): From (1.4) and (1.5), we have (see [4]). By (1.6), we easily get , where are the Bernoulli polynomials of degree . To give the new construction of the weighted -Bernoulli numbers and polynomials of higher order, we first use the multivariate -adic -integral on . The purpose of this paper is to give the higher-order -Bernoulli numbers and polynomials with weight and to derive a new explicit formulas by these numbers and polynomials. 2. On the Higher Order -Bernoulli Numbers with Weight For , we consider a sequence of -adic rational numbers as expansion of the weighted -Bernoulli numbers and polynomials of order as follows: From (2.1) and (2.2), we can derive the following equations: By (2.3) and (2.4), we get Therefore, by (2.5), we obtain the following theorem. Theorem 2.1. For , and , we have From (1.3) and (1.4), we note that Therefore, by (2.7), we obtain the following theorem. Theorem 2.2. For , we have From (2.2) and (2.3), we have Therefore, by (2.9), we obtain the following proposition. Proposition 2.3. For , we have By (2.2), we get where . Therefore, by (2.11), we obtain the following theorem. Theorem 2.4. For and , we have Let be a fixed integer. For , we set where satisfies the condition . Let be a primitive Dirichlet character with conductor . Then we consider the generalized -Bernoulli numbers with weight of order as follows: From (2.14), we have