`Discrete Dynamics in Nature and SocietyVolumeΒ 2011, Article IDΒ 176296, 11 pageshttp://dx.doi.org/10.1155/2011/176296`
Research Article

Some Relations between Twisted (β,π)-Euler Numbers with Weight Ξ± and π-Bernstein Polynomials with Weight Ξ±

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

Received 19 July 2011; Accepted 26 August 2011

Copyright Β© 2011 N. S. Jung 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

By using fermionic p-adic q-integral on , we give some interesting relationship between the twisted (h, q)-Euler numbers with weight Ξ± and the q-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 integers, denotes the field 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 (see [1β22]), be the locally constant space. For , we denote by the locally constant function . The -adic absolute value is defined by , where . In this paper, we assume that with as an indeterminate. The -number is defined by (see [1β22]). Note that . For the fermionic -adic -integral on is defined by Kim as follows: (see [1β7]). From (1.4), we note that where for .

For and , Kim defined -Bernstein polynomials, which are different -Bernstein polynomials of Phillips, as follows: (see [5]). In [9], the -adic extension of (1.6) is given by For , , , and with , twisted -Euler numbers with weight are defined by In the special case, , are called the -th twisted -Euler numbers with weight .

In this paper, we investigate some relations between the -Bernstein polynomials and the twisted -Euler numbers with weight . From these relations, we derive some interesting identities on the twisted -Euler numbers and polynomials with weight .

2. Twisted (β,π)-Euler Numbers and Polynomials with Weight πΌ

By using -adic -integral and (1.8), we obtain We set By (2.1) and (2.2), we have Since , we obtain

Therefore, we obtain the following theorem.

Theorem 2.1. For and , we have Furthermore, with usual convention about replacing with .
Let . Then we see that In the special case, , let .
By (2.1), we get From (2.3) and (2.7), we note that By (2.9), we get the following recurrence formula:

By (2.10) and Theorem 2.1, we obtain the following theorem.

Theorem 2.2. For and , we have with usual convention about replacing with .
By (2.4), Theorem 2.1, and Theorem 2.2, we have

Therefore, we obtain the following theorem.

Theorem 2.3. For , we have
By (2.8), we see that

Therefore, we obtain the following theorem.

Theorem 2.4. For , we have Let . By Theorems 2.3 and 2.4, we get
From (2.16), we have

Therefore, we obtain the following corollary.

Corollary 2.5. For , we have
Kim defined the -Bernstein polynomials with weight of degree as below.
For , the -adic -Bernstein polynomials with weight of degree are given by compare [5, 10, 22] By (2.19), we get the symmetry of -Bernstein polynomials as follows: see [8]. Thus, by Corollary 2.5, (2.19), and (2.20), we see that
For with , we have Let us take the fermionic -integral on for the -Bernstein polynomials with weight of degree as follows:

Therefore, by (2.22) and (2.23), we obtain the following theorem.

Theorem 2.6. Let with . Then we have Moreover,
Let with . Then we get

Therefore, by (2.26), we obtain the following theorem.

Theorem 2.7. For with , we have
From the binomial theorem, we can derive the following equation:

Thus, by (2.28) and Theorem 2.7, we obtain the following corollary.

Corollary 2.8. Let with . Then we have
For and with , let with . Then we take the fermionic -adic -integral on for the -Bernstein polynomials with weight of degree as follows:

Therefore, by (2.30), we obtain the following theorem.

Theorem 2.9. For with , let with . Then we get
By the definition of -Bernstein polynomials with weight and the binomial theorem, we easily get

Therefore, we have the following corollary.

Corollary 2.10. For with , let with . Then we have

References

1. L.-C. Jang, W.-J. Kim, and Y. Simsek, βA study on the p-adic integral representation on ${\mathrm{\beta €}}_{p}$ associated with Bernstein and Bernoulli polynomials,β Advances in Difference Equations, Article ID 163217, 6 pages, 2010.
2. 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. 15β27, 2007.
3. T. Kim, βSome identities on the q-Euler polynomials of higher order and q-Stirling numbers by the fermionic p-adic integral on ${\mathrm{\beta €}}_{p}$,β Russian Journal of Mathematical Physics, vol. 16, no. 4, pp. 484β491, 2009.
4. T. Kim, βBarnes-type multiple q-zeta functions and q-Euler polynomials,β Journal of Physics. A. Mathematical and Theoretical, vol. 43, no. 25, Article ID 255201, 11 pages, 2010.
5. T. Kim, βA note on q-Bernstein polynomials,β Russian Journal of Mathematical Physics, vol. 18, no. 1, pp. 73β82, 2011.
6. T. Kim, βq-Volkenborn integration,β Russian Journal of Mathematical Physics, vol. 9, no. 3, pp. 288β299, 2002.
7. T. Kim, βq-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients,β Russian Journal of Mathematical Physics, vol. 15, no. 1, pp. 51β57, 2008.
8. T. Kim, J. Choi, Y. H. Kim, and C. S. Ryoo, βOn 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.
9. T. Kim, B. Lee, J. Choi, and Y. H. Kim, βA new approach of q-Euler numbers and polynomials,β Proceedings of the Jangjeon Mathematical Society, vol. 14, no. 1, pp. 7β14, 2011.
10. T. Kim, J. Choi, and Y.-H. Kim, βSome identities on the q-Bernstein polynomials, q-Stirling numbers and q-Bernoulli numbers,β Advanced Studies in Contemporary Mathematics, vol. 20, no. 3, pp. 335β341, 2010.
11. T. Kim, βSome identities for the Bernoulli, the Euler and the Genocchi numbers and polynomials,β Advanced Studies in Contemporary Mathematics, vol. 20, no. 1, pp. 23β28, 2010.
12. T. Kim, βThe modified q-Euler numbers and polynomials,β Advanced Studies in Contemporary Mathematics, vol. 16, no. 2, pp. 161β170, 2008.
13. T. Kim, βNote on the Euler numbers and polynomials,β Advanced Studies in Contemporary Mathematics, vol. 17, no. 2, pp. 131β136, 2008.
14. H. Ozden and Y. Simsek, βA new extension of q-Euler numbers and polynomials related to their interpolation functions,β Applied Mathematics Letters, vol. 21, no. 9, pp. 934β939, 2008.
15. H. Ozden and Y. Simsek, βInterpolation function of the h, q-extension of twisted Euler numbers,β Computers & Mathematics with Applications, vol. 56, no. 4, pp. 898β908, 2008.
16. H. Ozden, Y. Simsek, and I. N. Cangul, βEuler polynomials associated with p-adic q-Euler measure,β General Mathematics, vol. 15, no. 2, pp. 24β37, 2007.
17. S.-H. Rim, J.-H. Jin, E.-J. Moon, and S.-J. Lee, βOn multiple interpolation functions of the q-Genocchi polynomials,β Journal of Inequalities and Applications, Article ID 351419, 13 pages, 2010.
18. S.-H. Rim, S. J. Lee, E. J. Moon, and J. H. Jin, βOn the q-Genocchi numbers and polynomials associated with q-zeta function,β Proceedings of the Jangjeon Mathematical Society, vol. 12, no. 3, pp. 261β267, 2009.
19. C. S. Ryoo, βOn the generalized Barnes type multiple q-Euler polynomials twisted by ramified roots of unity,β Proceedings of the Jangjeon Mathematical Society, vol. 13, no. 2, pp. 255β263, 2010.
20. C. S. Ryoo, βA note on the weighted q-Euler numbers and polynomials,β Advanced Studies in Contemporary Mathematics, vol. 21, pp. 47β54, 2011.
21. Y. Simsek, O. Yurekli, and V. Kurt, βOn interpolation functions of the twisted generalized Frobinuous-Euler numbers,β Advanced Studies in Contemporary Mathematics, vol. 14, pp. 49β68, 2007.
22. Y. Simsek and M. Acikgoz, βA 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.