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Research Article | Open Access

Volume 2008 |Article ID 295307 | https://doi.org/10.1155/2008/295307

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

Accepted18 Jun 2008
Published12 Oct 2008

#### Abstract

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.

#### 1. Introduction

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. [113]) for all . Hence, . For a fixed odd positive integer with , letwhere lies in . For any ,is known to be a distribution on (cf. [113]).

We say that is uniformly differentiable function at a point and denote this property by , if the difference quotientshave a limit as (cf. [2, 6, 7]). The -adic -integral of a function was defined as

By using -adic -integrals on , it is well known thatwhere . Then we note that the Bernoulli polynomials were defined asFrom (1.6) and (1.7), we havefor all . We note that .

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

Proof . From (1.5), we can derive (2.4) as follows: since andfor , and .
Let be the generating function of -Bernoulli polynomials as follows:From (2.2) and (2.7), we can obtain the following theorem.

Theorem 2.3. Let be as in the above generating function. Then, one has

Proof . By using (2.2) and (2.7), we can derive (2.8) as follows:

#### 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

#### Acknowledgment

This paper was supported by Konkuk University in 2008.

#### References

1. L. Carlitz, “$q$-Bernoulli numbers and polynomials,” Duke Mathematical Journal, vol. 15, no. 4, pp. 987–1000, 1948.
2. M. Cenkci, Y. Simsek, and V. Kurt, “Further remarks on multiple $p$-adic $q$-$L$-function of two variables,” Advanced Studies in Contemporary Mathematics (Kyungshang), vol. 14, no. 1, pp. 49–68, 2007. View at: Google Scholar | MathSciNet
3. T. Kim, “On explicit formulas of $p$-adic $q$-$L$-functions,” Kyushu Journal of Mathematics, vol. 48, no. 1, pp. 73–86, 1994.
4. T. Kim, “$q$-Volkenborn integration,” Russian Journal of Mathematical Physics, vol. 9, no. 3, pp. 288–299, 2002.
5. T. Kim, “On a $q$-analogue of the $p$-adic log gamma functions and related integrals,” Journal of Number Theory, vol. 76, no. 2, pp. 320–329, 1999.
6. T. Kim, L. C. Jang, and S. H. Rim, “An extension of $q$-zeta function,” International Journal of Mathematics and Mathematical Sciences, vol. 2004, no. 49, pp. 2649–2651, 2004.
7. T. Kim, “The modified $q$-Euler numbers and polynomials and polynomials,” Advanced Studies in Contemporary Mathematics, vol. 16, pp. 161–170, 2008. View at: Google Scholar
8. H. Ozden, Y. Simsek, S.-H. Rim, and I. N. Cangul, “A note on $p$-adic $q$-Euler measure,” Advanced Studies in Contemporary Mathematics (Kyungshang), vol. 14, no. 2, pp. 233–239, 2007. View at: Google Scholar | MathSciNet
9. T. Kim, “On $p$-adic $q$-$L$-functions and sums of powers,” Discrete Mathematics, vol. 252, no. 1–3, pp. 179–187, 2002.
10. T. Kim, “An invariant $p$-adic $q$-integrals on ${ℤ}_{p}$,” Applied Mathematics Letters, vol. 21, no. 2, pp. 105–108, 2008. View at: Publisher Site | Google Scholar
11. T. Kim, L. C. Jang, S.-H. Rim, and H.-K. Pak, “On the twisted $q$-zeta functions and $q$-Bernoulli polynomials,” Far East Journal of Applied Mathematics, vol. 13, no. 1, pp. 13–21, 2003.
12. K. Shriatani and S. Yamamoto, “On a $p$-adic interpolation function for the Euler numbers and its derivatives,” Memoirs of the Faculty of Science, Kyushu University, vol. 76, no. 2, pp. 320–329, 1999. View at: Google Scholar
13. Y. Simsek, “On $p$-adic twisted $q$-$L$-functions related to generalized twisted Bernoulli numbers,” Russian Journal of Mathematical Physics, vol. 13, no. 3, pp. 340–348, 2006. View at: Publisher Site | Google Scholar | MathSciNet

Copyright © 2008 Lee-Chae Jang. 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.