Discrete Dynamics in Nature and Society

Volume 2011 (2011), Article ID 487490, 7 pages

http://dx.doi.org/10.1155/2011/487490

## New Construction Weighted -Genocchi Numbers and Polynomials Related to Zeta Type Functions

^{1}Department of Mathematics, Faculty of Arts and Science, University of Gaziantep, 27310 Gaziantep, Turkey^{2}Department of Applied Mathematics, Pukyong National University, Busan 608-737, Republic of Korea

Received 31 March 2011; Revised 17 June 2011; Accepted 11 July 2011

Academic Editor: Guang Zhang

Copyright © 2011 Serkan Araci et al. 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.

#### Abstract

The fundamental aim of this paper is to construct -Genocchi numbers and polynomials with weight . We shall obtain some interesting relations by using -adic -integral on in the sense of fermionic. Also, we shall derive the -extensions of zeta type functions with weight from the Mellin transformation of this generating function which interpolates the -Genocchi numbers and polynomials with weight at negative integers.

#### 1. Introduction, Definitions, and Notations

Let be a fixed odd prime number. Throughout this paper we use the following notations. denotes the ring of -adic rational integers, denotes the field of rational numbers, denotes the field of -adic rational numbers, and denotes the completion of algebraic closure of . Let be the set of natural numbers and . The -adic absolute value is defined by . In this paper, we assume as an indeterminate. In [1–3], defined the fermionic -adic -integral on as follows:

is a -extension of which is defined by see [1–15].

Note that .

Let . By the definition (1.1) we easily get

Continuing this process, we obtain easily the relation

-Genocchi numbers are defined as follows: with the usual convention about replacing by (see [6]).

In this paper, we constructed -Genocchi numbers and polynomials with weight . By using fermionic -adic -integral equations on , we investigated some interesting identities and relations on the -Genocchi numbers and polynomials with weight . Furthermore, we derive the -extensions of zeta type functions with weight from the Mellin transformation of this generating function which interpolates the -Genocchi polynomials with weight .

#### 2. On the Weighted -Genocchi Numbers and Polynomials

In this section, by using fermionic -adic -integral equations on , some interesting identities and relation on the -Genocchi numbers and polynomials with weight are shown.

*Definition 2.1. *Let and . Then the -Genocchi numbers with weight defined by as follows:

If we take to (2.1), then we have, (see [5]).

From (2.1), we obtain

Therefore, we obtain the following theorem.

Theorem 2.2. *For and . Then
*

In (1.1), one takes ,

From [12], we obtain -Genocchi numbers with weight witt's type formula as follows.

Theorem 2.3. *For and . Then
*

From (2.1), one easily gets

By (2.6), one has

Therefore, we obtain the following corollary.

Corollary 2.4. *If . Let . Then
*

Now, one considers the -Genocchi polynomials with weight as follows:

From (2.9), one sees that Let . Then, one has

By (1.4), one sees that

Therefore, we obtain the following theorem.

Theorem 2.5. *For , and , one has
*

In (1.3), it is known that

If we take , then one has

Therefore, by (2.15), we obtain the following theorem.

Theorem 2.6. *For and , one has
*

From (2.9), one can easily derive the following:

Therefore, by (2.17), we obtain the following theorem.

Theorem 2.7. *For , and *

#### 3. Interpolation Function of the Polynomials

In this section, we give interpolation function of the generating functions of -Genocchi polynomials with weight . For and , by applying the Mellin transformation to (2.11), we obtain so we have

We define -extension zeta type function as follows.

Theorem 3.1. *For , , and . One has
** can be continued analytically to an entire function.*

By subsituting into (3.3) one easily gets We obtain the following theorem.

Theorem 3.2. *For and , . Then one defines
*

#### References

- T. Kim, “$q$-Euler numbers and polynomials associated with $p$-adic $q$-integrals,”
*Journal of Nonlinear Mathematical Physics*, vol. 14, no. 1, pp. 15–27, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - T. Kim, “New approach to $q$-Euler polynomials of higher order,”
*Russian Journal of Mathematical Physics*, vol. 17, no. 2, pp. 218–225, 2010. View at Publisher · View at Google Scholar - T. Kim, “Some identities on the $q$-Euler polynomials of higher order and $q$-Stirling numbers by the fermionic $p$-adic integral on ${\mathbb{Z}}_{p}$,”
*Russian Journal of Mathematical Physics*, vol. 16, no. 4, pp. 484–491, 2009. View at Publisher · View at Google Scholar - M. Acikgoz and Y. Simsek, “A new generating function of $q$-Bernstein-type polynomials and their interpolation function,”
*Abstract and Applied Analysis*, vol. 2010, Article ID 769095, 12 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. Araci, D. Erdal, and J. J. Seo, “A Study on the weighted $q$-Genocchi Numbers and Polynomials Their Interpolation function,” submitted.
- S. Araci, J. J. Seo, and D. Erdal, “Different Approach On The $(h,q)$ Genocchi Numbers and Polynomials Associated with $q$-Bernstein Polynomials,” submitted.
- T. Kim, “A new approach to $q$-Zeta function,”
*Advanced Studies in Contemporary Mathematics*, vol. 11, no. 2, pp. 157–162, 2005. View at Google Scholar · View at Zentralblatt MATH - T. Kim, “On the $q$-extension of Euler and Genocchi numbers,”
*Journal of Mathematical Analysis and Applications*, vol. 326, no. 2, pp. 1458–1465, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - T. Kim, “On the multiple $q$-Genocchi and Euler numbers,”
*Russian Journal of Mathematical Physics*, vol. 15, no. 4, pp. 481–486, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - T. Kim, “On the weighted $q$-Bernoulli numbers and polynomials,”
*Advanced Studies in Contemporary Mathematics*, vol. 21, no. 2, pp. 207–215, 2011. View at Google Scholar - T. Kim, “A note on the $q$-Genocchi numbers and polynomials,”
*Journal of Inequalities and Applications*, vol. 2007, Article ID 71452, 8 pages, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - T. Kim, “$q$-Volkenborn integration,”
*Russian Journal of Mathematical Physics*, vol. 9, no. 3, pp. 288–299, 2002. View at Google Scholar · View at Zentralblatt MATH - T. Kim, “An invariant $p$-adic $q$-integral on ${\mathbb{Z}}_{p}$,”
*Applied Mathematics Letters*, vol. 21, no. 2, pp. 105–108, 2008. View at Publisher · View at Google Scholar - T. Kim, “A note on the $q$-genocchi numbers and polynomials,”
*Journal of Inequalities and Applications*, vol. 2007, Article ID 71452, 8 pages, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - T. Kim, J. Choi, Y. H. Kim, and L. C. Jang, “On $p$
-Adic analogue of $q$-Bernstein polynomials and related integrals,”
*Discrete Dynamics in Nature and Society*, vol. 2010, Article ID 179430, 9 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH