Abstract and Applied AnalysisVolume 2011 (2011), Article ID 123483, 9 pagesdoi:10.1155/2011/123483

Research Article

Some Identities of the Twisted -Genocchi Numbers and Polynomials with Weight and -Bernstein Polynomials with Weight

H. Y. Lee, N. S. Jung, and C. S. Ryoo

Department of Mathematics, Hannam University, Daejeon 306-791, Republic of Korea

Received 7 July 2011; Accepted 22 August 2011

Academic Editor: John Rassias

Copyright © 2011 H. Y. Lee 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.

Recently mathematicians have studied some interesting relations between -Genocchi numbers, -Euler numbers, polynomials, Bernstein polynomials, and -Bernstein polynomials. In this paper, we give some interesting identities of the twisted -Genocchi numbers, polynomials, and -Bernstein polynomials with weighted .

1. Introduction

Throughout this paper, let be a fixed odd prime number. The symbols , , and denote the ring of -adic integers, the field of -adic rational numbers, and the completion of algebraic closure of . Let be the set of natural numbers and let . As a well-known definition, the -adic absolute value is given by , where with . When one talks of -extension, is variously considered as an indeterminate, a complex number , or a -adic number . In this paper we assume that with .

We assume that is the space of the uniformly differentiable function on . For , Kim defined the fermionic -adic -integral on as follows: For , let be translation. As a well known equation, by (1.1), we have compared [1–4]. Throughout this paper we use the notation: (cf. [1–16]). for any with in the present -adic case. To investigate relation of the twisted -Genocchi numbers and polynomials with weight and the Bernstein polynomials with weight , we will use useful property for as follows;

The twisted -Genocchi numbers and polynomials with weight are defined by the generating function as follows, respectively: In the special case, , are called the th twisted -Genocchi numbers with weight (see [9]).

Let be the cyclic group of order and let see [9, 12–15].

Kim defined the -Bernstein polynomials with weight of degree as follows: compare [4, 7].

In this paper, we investigate some properties for the twisted -Genocchi numbers and polynomials with weight . By using these properties, we give some interesting identities on the twisted -Genocchi polynomials with weight and -Bernstein polynomials with weight .

2. Some Identities on the Twisted -Genocchi Polynomials with Weight and -Bernstein Polynomials with Weight

From (1.8), we can derive the following recurrence formula for the twisted -Genocchi numbers with weight : with usual convention about replacing by .

By (1.5), we easily get By (2.4), we obtain the theorem below.

Theorem 2.1. Let . For , one has

By (2.1), (2.2), and (2.3) we note that

Therefore, by (2.6), we obtain the theorem below.

Theorem 2.2. For with , one has

By (1.6) and Theorem 2.2, Hence, we obtain the corollary below.

Corollary 2.3. For , one has

By fermionic integral on ,Theorems 2.1 and 2.2, we note that

Therefore, we have the theorem below.

Theorem 2.4. For with , one has

By (1.4), Theorem 2.4, we take the fermionic -adic invariant integral on for one q-Bernstein polynomials as follows:

By symmetry of -Bernstein polynomials with weight of degree , we get the following formula;

Therefore, by (2.12) and (2.13), we have the theorem below.

Theorem 2.5. For with , one has Also, we note that

Therefore, we have the theorem below.

Theorem 2.6. For with , one has

By (2.11) and Theorem 2.6, we have the theorem below.

Theorem 2.7. Let with . Then one has

Let with . Then we get

Therefore, we obtain the theorem below.

Theorem 2.8. For , one has

By simple calculation, we easily see that Therefore, by (2.20) and Theorem 2.8, we obtain the theorem below.

Theorem 2.9. Let with . Then one has

For , , and let , then by the symmetry of -Bernstein polynomials with weight , we see that

Therefore, we have the theorem below.

Theorem 2.10. For with , one has where .

In the same manner as in (2.15), we can get the following relation: where with .

By Theorem 2.10 and (2.13), we have the following corollary.

Corollary 2.11. Let . For with , one has where .

References

T. Kim, “Some identities on the -Euler polynomials of higher order and -Stirling numbers by the fermionic -adic integral on ,” Russian Journal of Mathematical Physics, vol. 16, no. 4, pp. 484–491, 2009.

T. Kim, “Note on the Euler numbers and polynomials,” Advanced Studies in Contemporary Mathematics, vol. 17, no. 2, pp. 131–136, 2008.

T. Kim, “-Volkenborn integration,” Russian Journal of Mathematical Physics, vol. 9, no. 3, pp. 288–299, 2002.

T. Kim, J. Choi, and Y.-H. Kim, “Some identities on the -Bernstein polynomials, -Stirling numbers and -Bernoulli numbers,” Advanced Studies in Contemporary Mathematics, vol. 20, no. 3, pp. 335–341, 2010.

I. N. Cangul, H. Ozden, and Y. Simsek, “A new approach to -Genocchi numbers and their interpolation functions,” Nonlinear Analysis, vol. 71, no. 12, pp. e793–e799, 2009.

I. N. Cangul, H. Ozden, V. Kurt, and Y. Simsek, “On the higher-order --Genocchi numbers,” Advanced Studies in Contemporary Mathematics, vol. 19, no. 1, pp. 39–57, 2009.

T. Kim, L. C. Jang, and H. Yi, “A note on the modified -Bernstein polynomials,” Discrete Dynamics in Nature and Society, vol. 2010, Article ID 706483, 12 pages, 2010.

T. Kim, J. Choi, Y. H. Kim, and C. S. Ryoo, “On the fermionic -adic integral representation of Bernstein polynomials associated with Euler numbers and polynomials,” Journal of Inequalities and Applications, vol. 2010, Article ID 864247, 12 pages, 2010.

H. Y. Lee, “A note on the twisted -Genocchi numbers and polynomials with weight α,” Journal Of Applied Mathematics and Informatics. In press

H. Ozden, I. N. Cangul, and Y. Simsek, “Hurwitz type multiple genocchi Zeta function,” in Numerical Analysis and Applied Math, AIP Conference Proceedings, pp. 1148–1781, 2009.

H. Ozden, Y. Simsek, and H. M. Srivastava, “A unified presentation of the generating functions of the generalized Bernoulli, Euler and Genocchi polynomials,” Computers and Mathematics with Applications, vol. 60, no. 10, pp. 2779–2787, 2010.

S. H. Rim, J. H. Jin, E. J. Moon, and S. J. Lee, “Some identities on the -Genocchi polynomials of higher-order and -Stirling numbers by the fermionic -adic integral on ,” International Journal of Mathematics and Mathematical Sciences, vol. 2010, Article ID 860280, 14 pages, 2010.

C. S. Ryoo, “Some identities of the twisted -Euler numbers and polynomials associated with -Bernstein polynomials,” vol. 14, no. 2, pp. 239–248, 2011.

C. S. Ryoo, “Some relations between twisted -Euler numbers and Bernstein polynomials,” Advanced Studies in Contemporary Mathematics, vol. 21, no. 2, pp. 217–223, 2011.

Y. Simsek, V. Kurt, and D. Kim, “New approach to the complete sum of products of the twisted ()-Bernoulli numbers and polynomials,” Journal of Nonlinear Mathematical Physics, vol. 14, no. 1, pp. 44–56, 2007.

Y. Simsek and M. Acikgoz, “A new generating function of -Bernstein-type polynomials and their interpolation function,” Abstract and Applied Analysis, vol. 2010, Article ID 769095, 12 pages, 2010.