Journal of Applied Mathematics

Volume 2011 (2011), Article ID 609054, 10 pages

http://dx.doi.org/10.1155/2011/609054

## A Note on Some Properties of the Weighted -Genocchi Numbers and Polynomials

Department of Mathematics and Computer Science, Konkuk University, Chungju 280-701, Republic of Korea

Received 30 July 2011; Revised 23 September 2011; Accepted 23 September 2011

Academic Editor: Mark A. Petersen

Copyright © 2011 L. C. 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.

#### Abstract

We consider the weighted -Genocchi numbers and polynomials. From the construction of the weighted -Genocchi numbers and polynomials, we investigate many interesting identities and relations satisfied by these new numbers and polynomials.

#### 1. Introduction

Let be a fixed odd prime number. Throughout this paper, , , , and , will, respectively, denote the ring of -adic integers, the field, of -adic rational numbers, the complex number field and the completion of algebraic closure of . Let be the normalized exponential valuation of such that (see [1–16]).

As well-known definition, the Euler numbers and Genocchi numbers are defined by with the usual convention of replacing by and with the usual convention of replacing by . We assume that with and that the -number of is defined by (see [1–19]).

In [9], Kim introduced ordinary fermionic -adic integral on , and he studied some interesting relations and identities related to -extension of Euler numbers and polynomials. In [8], he also introduced the -extension of the ordinary fermionic -adic integral on and he investigated many physical properties related to -Euler numbers and polynomials. Recently, Kim firstly introduced the meaning of the weighted -Euler numbers and polynomials associated with the weighted -Bernstein polynomials by using the fermionic invariant -adic integral on (see [14, 15]). In [16], Ryoo tried to study the weighted -Euler number and polynomials by the same method of Kim et al. in [14] and the -extension of the fermionic -adic invariant integrals on . As well-known properties, the Genocchi numbers are integers. The first few Genocchi numbers for are . The first few prime Genocchi numbers are −3 and 17, which occur for and 8. There are no others with . These properties are very important to study in the area of fermionic distribution and -adic numbers theory. By this reason, many mathematicians and physicians have studied Genocchi and Euler numbers which are in the different areas. By the same motivation, we consider weighted -Genocchi polynomials and numbers by using the fermionic -adic -integral on which are constructed by Kim and Ryoo (cf. [8, 16]).

In this paper, we consider the -Genocchi numbers and polynomials with weighted . From the construction of the weighted -Genocchi numbers and polynomials, we investigate many interesting identities and relations satisfied by these new numbers and polynomials.

#### 2. The Weighted -Genocchi Numbers and Polynomials

Let be the space of uniformly differentiable functions and, for , the fermionic -adic invariant integral of on is defined by Kim as follows: (see [1–16]). If we take , then we get By (1.2) and (2.2), we get From (2.3), For , the fermionic -adic -integral of on is defined by Kim as follows: (see [1–16]). From (2.5), we note that where and .

For , we consider the following fermionic -adic -integral on : where are called the th -Genocchi numbers with weight . From (2.7), we get By comparing the coefficients on the both sides of (2.7) and (2.8), we get From (2.9), we obtain the following theorem.

Theorem 2.1. *For and , one has
*

By the definition of fermionic -adic -integrals, we get Therefore, we obtain the following theorem.

Theorem 2.2. *For and , we have
*

By Theorem 2.2, we have the generating function of as follows: Let be the generating function of . Then, by (2.9) and (2.13), we get

The -Genocchi polynomials with weight are defined by From (2.15), we get By (2.15) and (2.16), we obtain the following theorem.

Theorem 2.3. *For and , one has
*

We note that From (2.17) and (2.18), we obtain the following theorem.

Theorem 2.4. *For and , one has
*

From (2.15), we note that Therefore, we obtain the following theorem.

Theorem 2.5. *For , one has
*

From (2.15) and (2.21), we obtain that Therefore, we obtain the following theorem.

Theorem 2.6. *For and , one has
*

From (2.6), if we take , then we get By (2.17) and (2.24), we obtain the following theorem.

Theorem 2.7. *For , and , one has
*

We remark that if we take in Theorem 2.7, then we have and if we take in Theorem 2.7, then we have From (2.27) with , we obtain the following corollary.

Corollary 2.8. *For and , one has
*

From (2.19), we note that From (2.29), we get with the usual convention about replacing by . By (2.28) and (2.30), we get From (2.28) and (2.31), we obtain the following theorem.

Theorem 2.9. *For and , one has
*

#### Acknowledgment

This paper was supported by the Konkuk University in 2011.

#### References

- I. N. Cangul, V. Kurt, H. Ozden, and Y. Simsek, “On the higher-order
*w*-*q*-Genocchi numbers,”*Advanced Studies in Contemporary Mathematics*, vol. 19, no. 1, pp. 39–57, 2009. View at Google Scholar - L.-C. Jang, “On multiple generalized
*w*-Genocchi polynomials and their applications,”*Mathematical Problems in Engineering*, vol. 2010, Article ID 316870, 8 pages, 2010. View at Publisher · View at Google Scholar - L.-C. Jang, “A new
*q*-analogue of Bernoulli polynomials associated with*p*-adic*q*-integrals,”*Abstract and Applied Analysis*, vol. 2008, Article ID 295307, 6 pages, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - L. Jang and T. Kim, “
*q*-Genocchi numbers and polynomials associated with fermionic*p*-adic invariant integrals on ${\mathbb{Z}}_{p}$,”*Abstract and Applied Analysis*, vol. 2008, Article ID 232187, 8 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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, “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, “New approach to
*q*-Euler, Genocchi numbers and their interpolation functions,”*Advanced Studies in Contemporary Mathematics*, vol. 18, no. 2, pp. 105–112, 2009. View at Google Scholar - T. Kim, “Barnes-type multiple
*q*-zeta functions and*q*-Euler polynomials,”*Journal of Physics A*, vol. 43, no. 25, Article ID 255201, 11 pages, 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 · View at MathSciNet - 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 · View at MathSciNet - T. Kim and B. Lee, “Some identities of the Frobenius-Euler polynomials,”
*Abstract and Applied Analysis*, vol. 2009, Article ID 639439, 7 pages, 2009. View at Publisher · View at Google Scholar - T. Kim, L.-C. Jang, and H. Yi, “A note on the modified
*q*-Bernstein polynomials,”*Discrete Dynamics in Nature and Society*, vol. 2010, Article ID 706483, 12 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - T. Kim, J. Choi, and Y. H. Kim, “
*q*-Bernstein polynomials associated with*q*-Stirling numbers and Carlitz's*q*-Bernoulli numbers,”*Abstract and Applied Analysis*, vol. 2010, Article ID 150975, 11 pages, 2010. View at Publisher · View at Google Scholar - T. Kim, B. Lee, J. Choi, Y. H. Kim, and S. H. Rim, “On the
*q*-Euler numbers and weighted q-bernstein polynomials,”*Advanced Studies in Contemporary Mathematics*, vol. 21, no. 1, pp. 13–18, 2011. View at Google Scholar - T. Kim, “On the weighted
*q*-Bernoulli numbers and polynomials,”*Advanced Studies in Contemporary Mathematics (Kyungshang)*, vol. 21, no. 2, pp. 207–215, 2011. View at Google Scholar - C. S. Ryoo, “A note on the weighted
*q*-Euler numbers and polynomials,”*Advanced Studies in Contemporary Mathematics*, vol. 21, no. 1, pp. 47–54, 2011. View at Google Scholar - L. Carlitz, “
*q*-Bernoulli numbers and polynomials,”*Duke Mathematical Journal*, vol. 15, pp. 987–1000, 1948. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. A. Kupershmidt, “Reflection symmetries of
*q*-Bernoulli polynomials,”*Journal of Nonlinear Mathematical Physics*, vol. 12, pp. 412–422, 2005. View at Publisher · View at Google Scholar · View at MathSciNet - V. Kurt, “A further symmetric relation on the analogue of the Apostol-Bernoulli and the analogue of the Apostol-Genocchi polynomials,”
*Applied Mathematical Sciences*, vol. 3, no. 53–56, pp. 2757–2764, 2009. View at Google Scholar