Abstract and Applied Analysis

VolumeΒ 2011, Article IDΒ 649248, 10 pages

http://dx.doi.org/10.1155/2011/649248

## A Study on the Fermionic -Adic -Integral Representation on Associated with Weighted -Bernstein and -Genocchi Polynomials

^{1}Department of Mathematics, Faculty of Science and Arts, 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 25 June 2011; Accepted 18 July 2011

Academic Editor: TokaΒ Diagana

Copyright Β© 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

We consider weighted -Genocchi numbers and polynomials. We investigated some interesting properties of the weighted -Genocchi numbers related to weighted -Bernstein polynomials by using fermionic -adic integrals on .

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

The main motivation of this paper is [1] by Kim, in which he introduced and studied properties of -Bernoulli numbers and polynomials with weight . Recently, many mathematicians have studied weighted special polynomials (see [1β5]).

This numbers and polynomials are used in not only number theory, complex analysis, and the other branch of mathematics, but also in other parts of the -adic analysis and mathematical physics. Kurt Hensel (1861β1941) invented the so-called -adic numbers around the end of the nineteenth century. In spite of their being already one hundred years old, these numbers are still today enveloped in an aura of mystery within scientific community [6] although they have penetrated several mathematical fields such as number theory, algebraic geometry, algebraic topology, analysis, and mathematical physics (see, for details, [6β8]).

The -adic -integral (or -Volkenborn integral) are originally constructed by Kim [9]. The -Volkenborn integral is used in mathematical physics, for example, the functional equation of the -zeta function, the -Stirling numbers, and -Mahler theory of integration with respect to the ring together with Iwasawa's -adic - function.

Let be a fixed odd prime number. Throughout this paper, we use the following notations. By , we denote 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 [10β12], let UD be the space of uniformly differentiable functions on . For UD , the fermionic -adic -integral on is defined by Kim

For and , Kim et al. defined weighted -Bernstein polynomials as follows: (see [13, 14]). When we put and in (1.2), , and we obtain the classical Bernstein polynomials (see [13, 14]), where is a -extension of which is defined by (see [1β4, 7, 9β12, 14β26]). Note that .

In [3], For , S. Araci et al. defined weighted -Genocchi polynomials as follows:

In the special case, , ββare called the -Genocchi numbers with weight .

In [3], For and , S. Araci et al. defined -Genocchi numbers with weight as follows:

In this paper we obtained some relations between the weighted -Bernstein polynomials and the -Genocchi numbers. From these relations, we derive some interesting identities on the -Genocchi numbers and polynomials with weight .

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

By the definition of -Genocchi polynomials with weight , we easily get

Therefore, we obtain the following theorem.

Theorem 2.1. *For , one has
**
with usual convention about replacing by .*

By Theorem 2.1, we have

By (1.4), we get

Therefore, we obtain the following theorem.

Theorem 2.2. *For , one has
*

From (1.5) and Theorem 2.1, we have the following theorem.

Theorem 2.3. *For , one has
**
with usual convention about replacing by .*

For , by Theorem 2.3, we note that

Therefore, we have the following theorem.

Theorem 2.4. *For , one has
*

From Theorem 2.2 and (2.5), we see that

Therefore, we get the following theorem.

Theorem 2.5. *For , one has
*

Let . By Theorems 2.4 and 2.5, we get

From (2.11), we get the following corollary.

Corollary 2.6. *For , one has
*

#### 3. Novel Identities on the Weighted -Genocchi Numbers

In this section, we derive concerning the some interesting properties of -Genocchi numbers via the -adic -integral on , in the sense of fermionic and weighted -Bernstein polynomials.

By (3.1), Kim et al. get the symmetry of -Bernstein polynomials weighted as follows: (see [4]). Thus, from Corollary 2.6, (3.1), and (3.2), we see that

For and with , we obtain

Let us take the fermionic -adic -integral on for the weighted -Bernstein polynomials of degree as follows:

Therefore, by (3.4) and (3.5), we obtain the following theorem.

Theorem 3.1. *For and with , one has
*

Let and with . βThen, we get

Therefore, we obtain the following theorem.

Theorem 3.2. *For and with , one has
*

From the binomial theorem, we can derive

Thus, for Theorem 3.4 and (3.13), we can obtain the following corollary.

Corollary 3.3. *For and with , one has
*

For and with , let and with . Then, we take the fermionic -adic -integral on for the weighted -Bernstein polynomials of degree as follows:

Therefore, we obtain the following theorem.

Theorem 3.4. *For with , let and with . Then, one has
*

From the definition of weighted -Bernstein polynomials and the binomial theorem, we easily get

Therefore, from (3.13) and Theorem 3.4, we have the following corollary.

Corollary 3.5. *For with , let and with . One has
*

#### Acknowledgments

The authors wish to express their sincere gratitude to the referees for their valuable suggestions and comments and Professor Toka Diagana for his cooperation and help.

#### References

- T. Kim, βOn the weighted
*q*-Bernoulli numbers and polynomials,β*Advanced Studies in Contemporary Mathematics*, vol. 21, no. 2, pp. 207β215, 2011. View at Google Scholar - S. Araci, J.-J. Seo, and D. Erdal, βNew construction weighted (
*h, q*)-Genocchi numbers and polynomials related to Zeta type function,β*Discrete Dynamics in Nature and Society*, vol. 2011, Article ID 487490, p. 7, 2011. View at Publisher Β· View at Google Scholar - S. Araci, M. Acikgoz, and J.-J. Seo, βA study on the weighted
*q*-Genocchi numbers and polynomials with their interpolation function,β submitted to*Abstract and Applied Analysis*. - T. Kim, A. Bayad, and Y.-H. Kim, βA study on the
*p*-adic*q*-integral representation on ${\mathbb{Z}}_{p}$ associated with the weighted*q*-Bernstein and*q*-Bernoulli polynomials,β*Journal of Inequalities and Applications*, vol. 2011, Article ID 513821, 8 pages, 2011. View at Publisher Β· View at Google Scholar - C. S. Ryoo, βA note on the weighted
*q*-Euler numbers and polynomials,β*Advanced Studies in Contemporary Mathematics*, vol. 21, no. 1, pp. 47β54, 2011. View at Google Scholar - A. M. Robert,
*A course in p-adic analysis*, Springer, New York, NY, USA, 2000. - T. Kim, β
*q*-Volkenborn integration,β*Russian Journal of Mathematical Physics*, vol. 9, no. 3, pp. 288β299, 2002. View at Google Scholar Β· View at Zentralblatt MATH - N. Koblitz,
*p-Adic Analysis: A Short Course on Recent Work*, vol. 46 of*London Mathematical Society Lecture Note Series*, Cambridge University Press, Cambridge, UK, 1980. - T. Kim, βOn a
*q*-analogue of the*p*-adic log gamma functions and related integrals,β*Journal of Number Theory*, vol. 76, no. 2, pp. 320β329, 1999. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - T. Kim, J. Choi, Y. H. Kim, and L. C. Jang, βOn
*p*-adic analogue of*q*-Bernstein polynomials and related integrals,β*Discrete Dynamics in Nature and Society*, vol. 2010, Article ID 179430, 9 pages, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - T. Kim, β
*q*-Euler numbers and polynomials associated with*p*-adic*q*-integrals,β*Journal of Nonlinear Mathematical Physics*, vol. 14, no. 1, pp. 15β27, 2007. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - T. Kim, βNew approach to
*q*-Euler polynomials of higher order,β*Russian Journal of Mathematical Physics*, vol. 17, no. 2, pp. 218β225, 2010. View at Publisher Β· View at Google Scholar - M. Açıkgöz and S. Araci, βA study on the integral of the product of several type Bernstein polynomials,β
*IST Transaction of Applied Mathematics-Modelling and Simulation*, vol. 1, no. 1, pp. 10–β14, 2010. View at Google Scholar - M. Açıkgöz and Y. Şimşek, βA new generating function of
*q*-Bernstein type polynomials and their interpolation function,β*Abstract and Applied Analysis*, vol. 2010, Article ID 769095, 12 pages, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - S. Araci, D. Erdal, and D.-J. Kang, βSome new properties on the
*q*-Genocchi numbers and polynomials associated with*q*-Bernstein polynomials,β*Honam Mathematical Journal*, vol. 33, no. 2, pp. 261β270, 2011. View at Google Scholar - S. Araci, N. Aslan, and J.-J. Seo, βA note on the weighted twisted Dirichlet's type
*q*-Euler numbers and polynomials,β*Honam Mathematical Journal*. In press. - T. Kim, βOn the
*q*-extension of Euler and Genocchi numbers,β*Journal of Mathematical Analysis and Applications*, vol. 326, no. 2, pp. 1458β1465, 2007. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - T. Kim, βOn the multiple
*q*-Genocchi and Euler numbers,β*Russian Journal of Mathematical Physics*, vol. 15, no. 4, pp. 481β486, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - T. Kim, βA note on the
*q*-Genocchi numbers and polynomials,β*Journal of Inequalities and Applications*, vol. 2007, Article ID 71452, 8 pages, 2007. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - T. Kim, βA note on
*q*-Bernstein polynomials,β*Russian Journal of Mathematical Physics*, vol. 18, no. 1, pp. 73β82, 2011. View at Publisher Β· View at Google Scholar - T. Kim, β
*q*-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients,β*Russian Journal of Mathematical Physics*, vol. 15, no. 1, pp. 51β57, 2008. View at Google Scholar Β· View at Zentralblatt MATH - T. Kim, J. Choi, Y. H. Kim, and C. S. Ryoo, βOn the fermionic
*p*-adic integral representation of Bernstein polynomials associated with Euler numbers and polynomials,β*Journal of Inequalities and Applications*, vol. 2010, Article ID 864247, 12 pages, 2010. View at Google Scholar - T. Kim, J. Choi, and Y. H. Kim, βSome identities on the
*q*-Bernstein polynomials,*q*-Stirling numbers and*q*-Bernoulli numbers,β*Advanced Studies in Contemporary Mathematics*, vol. 20, no. 3, pp. 335β341, 2010. View at Google Scholar - T. Kim, βAn invariant
*p*-adic*q*-integral on ${\mathbb{Z}}_{p}$,β*Applied Mathematics Letters*, vol. 21, no. 2, pp. 105β108, 2008. View at Publisher Β· View at Google Scholar - T. Kim, J. Choi, and Y. H. Kim, β
*q*-Bernstein polynomials associated with*q*-Stirling numbers and Carlitz's*q*-Bernoulli numbers,β*Abstract and Applied Analysis*, vol. 2010, Article ID 150975, 11 pages, 2010. View at Publisher Β· View at Google Scholar - T. Kim, βSome identities on the
*q*-Euler polynomials of higher order and*q*-Stirling numbers by the fermionic*p*-adic integral on ${\mathbb{Z}}_{p}$,β*Russian Journal of Mathematical Physics*, vol. 16, no. 4, pp. 484β491, 2009. View at Publisher Β· View at Google Scholar Β· View at MathSciNet