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International Journal of Mathematics and Mathematical Sciences
Volume 2011, Article ID 482840, 8 pages
http://dx.doi.org/10.1155/2011/482840
Research Article

Some Identities on the Twisted -Genocchi Numbers and Polynomials Associated with -Bernstein Polynomials

1Department of Mathematics Education, Kyungpook National University, Daegu 702-701, Republic of Korea
2Department of Mathematics, Kyungpook National University, Daegu 702-701, Republic of Korea

Received 9 October 2011; Accepted 15 November 2011

Academic Editor: Taekyun Kim

Copyright © 2011 Seog-Hoon Rim and Sun-Jung Lee. 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 give some interesting identities on the twisted ()-Genocchi numbers and polynomials associated with -Bernstein polynomials.

1. Introduction

Let be a fixed odd prime number. Throughout this paper, we always make use of the following notations: denotes the ring of rational integers, denotes the ring of -adic rational integer, denotes the ring of -adic rational numbers, and denotes the completion of algebraic closure of , respectively. Let be the set of natural numbers and . Let be the cyclic group of order and let The -adic absolute value is defined by , where ( and with () = () = () = 1). In this paper we assume that with as an indeterminate.

The -number is defined by (see [115]). Note that . Let be the space of uniformly differentiable function on . For , Kim defined the fermionic -adic -integral on as follows: (see [26, 815]). From (1.3), we note that (see [46, 812]), where for . For and , Kim defined the -Bernstein polynomials of the degree as follows: (see [1315]). For and , let us consider the twisted ()-Genocchi polynomials as follows: Then, is called th twisted ()-Genocchi polynomials.

In the special case, and are called the th twisted ()-Genocchi numbers.

In this paper, we give the fermionic -adic integral representation of -Bernstein polynomial, which are defined by Kim [13], associated with twisted ()-Genocchi numbers and polynomials. And we construct some interesting properties of -Bernstein polynomials associated with twisted ()-Genocchi numbers and polynomials.

2. On the Twisted -Genocchi Numbers and Polynomials

From (1.6), we note that We also have Therefore, we obtain the following theorem.

Theorem 2.1. For and , one has with usual convention about replacing by .
By (1.6) and (2.1) one gets

Therefore, we obtain the following theorem.

Theorem 2.2. For and , one has
From (1.5), one gets the following recurrence formula:

Therefore, we obtain the following theorem.

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

From Theorem 2.3, we note that

Therefore, we obtain the following theorem.

Theorem 2.4. For and , one has

Remark 2.5. We note that Theorem 2.4 also can be proved by using fermionic integral equation (1.4) in case of .

By (2.4) and Theorem 2.2, we get

Therefore, we obtain the following theorem.

Theorem 2.6. For and , one has

Let . By Theorems 2.4 and 2.6, we get Therefore, we obtain the following corollary.

Corollary 2.7. For and , one has
By (1.5), we get the symmetry of -Bernstein polynomials as follows: (see [11]).

Thus, by Corollary 2.7 and (2.14), we get

From (2.15), we have the following theorem.

Theorem 2.8. For and , one has
For with , fermionic -adic invariant integral for multiplication of two -Bernstein polynomials on can be given by the following:

From Theorem 2.8 and (2.17), we have the following corollary.

Corollary 2.9. For and , one has
Let with . Then we get

From (2.19), we have the following theorem.

Theorem 2.10. For and , one has
Let with , fermionic -adic invariant integral for multiplication of two -Bernstein polynomials on can be given by the following:

From Theorem 2.10 and (2.21), we have the following corollary.

Corollary 2.11. For and , one has

Acknowledgment

The authors would like to thank the anonymous referee for his/her excellent detail comments and suggestion.

References

  1. 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
  2. T. Kim, “A note on p-adic q-integral on p Associated with q-Euler numbers,,” Advanced Studies in Contemporary Mathematics, vol. 15, no. 2, pp. 133–138, 2007. View at Google Scholar
  3. T. Kim, “A note on q-Volkenborn integration,” Proceedings of the Jangjeon Mathematical Society, vol. 8, no. 1, pp. 13–17, 2005. View at Google Scholar
  4. 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 · View at MathSciNet
  5. 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
  6. T. Kim, “Some identities on the q-Euler polynomials of higher order and q-Stirling numbers by the fermionic p-adic integral on p,” Russian Journal of Mathematical Physics, vol. 16, no. 4, pp. 484–491, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  7. T. Kim, “Symmetry of power sum polynomials and multivariate fermionic p-adic invariant integral on p,” Russian Journal of Mathematical Physics, vol. 16, no. 1, pp. 93–96, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  8. 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 · View at MathSciNet
  9. H. M. Srivastava, T. Kim, and Y. Simsek, “q-Bernoulli numbers and polynomials associated with multiple q-zeta functions and basic L-series,” Russian Journal of Mathematical Physics, vol. 12, no. 2, pp. 241–268, 2005. View at Google Scholar
  10. 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
  11. R. Dere and Y. Simsek, “Genocchi polynomials associated with the Umbral algebra,” Applied Mathematics and Computation, vol. 218, no. 3, pp. 756–761, 2011. View at Publisher · View at Google Scholar
  12. H. Ozden, I. N. Cangul, and Y. Simsek, “A new approach to q-Genocchi numbers and their interpolation functions,” Nonlinear Analysis, vol. 71, no. 12, pp. e793–e799, 2009. View at Publisher · View at Google Scholar
  13. T. Kim, “A note on q-Bernstein polynomials,” Russian Journal of Mathematical Physics, vol. 18, no. 1, pp. 73–82, 2011. View at Publisher · View at Google Scholar
  14. L. C. Jang, W. J. Kim, and Y. Simsek, “A study on the p-adic integral representation on p associated with Bernstein and Bernoulli polynomials,” Advances in Difference Equations, vol. 2010, Article ID 163217, 6 pages, 2010. View at Publisher · View at Google Scholar
  15. D. V. Dolgy, D. J. Kang, T. Kim, and B. Lee, “Some new identities on the twisted (h, q)-Euler numbers and q-Bernstein polynomials,” Journal of Computational Analysis and Applications. In press.