Research Article | Open Access
Lee-Chae Jang, "A New -Analogue of Bernoulli Polynomials Associated with -Adic -Integrals", Abstract and Applied Analysis, vol. 2008, Article ID 295307, 6 pages, 2008. https://doi.org/10.1155/2008/295307
A New -Analogue of Bernoulli Polynomials Associated with -Adic -Integrals
We will study a new -analogue of Bernoulli polynomials associated with -adic -integrals. Furthermore, we examine the Hurwitz-type -zeta functions, replacing -adic rational integers with a -analogue for a -adic number with , which interpolate -analogue of Bernoulli polynomials.
Let be a fixed odd prime number. Throughout this paper , , , and will, respectively, represent the ring of -adic rational integers, the field of -adic rational numbers, the complex number field, and the -adic completion of the algebraic closure of . The -adic absolute value in is normalized so that and is a -adic number in with . We use the notation(cf. [1–13]) for all . Hence, . For a fixed odd positive integer with , letwhere lies in . For any ,is known to be a distribution on (cf. [1–13]).
In Section 2, we study a -analogue of Bernoulli polynomials associated with -adic -integrals—simply, we say -Bernoulli polynomials. In Section 3, we examine Hurwitz-type -zeta functions, replacing -adic rational integers with a -analogue for a -adic number with , which interpolate -analogue of Bernoulli polynomials.
2. A New -Analogue of Bernoulli Polynomials
In this section, from the view of (1.8), we can define a new -analogue of Bernoulli polynomials as follows:We note that are called the -Bernoulli numbers. Then we find some properties of -Bernoulli numbers and polynomials as follows.
Theorem 2.1. For , one has
Proof . From (1.5) with , we can find the following:
Theorem 2.2. For and being an odd positive integer with , one has
Theorem 2.3. Let be as in the above generating function. Then, one has
3. A New Formula for Hurwitz-Type -Zeta Functions
In this section, we consider the generating functions which interpolate the -Bernoulli polynomials as follows:From (3.1), we directly obtain the following theorem.
Theorem 3.1. For each , one has
Proof . By the th differentiation on both sides of (3.1), we
can derive (3.2) as follows:
We remark thatfor . From (3.2), we derive a -extension of Hurwitz-type zeta function as follows: for with and , we defineNote that the functions are analytic on and they have simple pole at . From (3.2), (3.4), and (3.5), we can see that Hurwitz-type -zeta functions interpolate -Bernoulli polynomials as follows.
Theorem 3.2. For each , one has
This paper was supported by Konkuk University in 2008.
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