Table of Contents Author Guidelines Submit a Manuscript
International Journal of Mathematics and Mathematical Sciences
Volume 2010, Article ID 860280, 14 pages
http://dx.doi.org/10.1155/2010/860280
Research Article

Some Identities on the -Genocchi Polynomials of Higher-Order and -Stirling Numbers by the Fermionic -Adic Integral on

Department of Mathematics, Kyungpook National University, Tagegu 702-701, Republic of Korea

Received 25 September 2010; Accepted 8 November 2010

Academic Editor: H. Srivastava

Copyright © 2010 Seog-Hoon Rim 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

A systemic study of some families of -Genocchi numbers and families of polynomials of Nörlund type is presented by using the multivariate fermionic -adic integral on . The study of these higher-order -Genocchi numbers and polynomials yields an interesting -analog of identities for Stirling numbers.

1. Introduction

Let be a fixed odd prime number. Throughout this paper, , and denote the ring of -adic rational integers, the field of -adic rational numbers, the complex number field, and the completion of the algebraic closure of , respectively. Let be the set of natural numbers and . Let be the normalized exponential valuation of with .

When one talks of -extension, is variously considered as an indeterminate, a complex , or a -adic number . If , then one normally assumes . If , then we assume . In this paper, we use the following notation: see [110]. Hence for all .

The -factorial is defined as , and the Gaussian binomial coefficient is defined by the standard rule (see [7, 9]). Note that . It readily follows from (1.2) that (see [4, 7]).

The -binomial formulas are known,

We say that is uniformly differentiable function at a point , and we write , if the difference quotients such that have a limit as . For , the -deformed fermionic -adic integral is defined as (see [7, 9]). Note that For , write . Then, we have Using (1.7), we can readily derive the Genocchi polynomials, , namely, (see [127]). Note that are referred to as the th Genocchi numbers. Let us now introduce the Genocchi polynomials of Nörlund type as follows: (see [7, 9]). In the special case , and are referred to as the Genocchi numbers of Nörlund type. Let be the shift operator. Then, the -difference operator is defined as (see [4, 7, 9]). It follows from (1.11) that where (see [5, 6, 10]). The -Stirling number of the second kind (as defined by Carlitz) is given by (see [7, 10]). By (1.12) and (1.13), we see that (see [6, 10]).

In this paper, the -extensions of (1.9) are considered in several ways. Using these -extensions, we derive some interesting identities and relations for Genocchi polynomials and numbers of Nörlund type. The purpose of this paper is to present a systemic study of some families -Genocchi numbers and polynomials of Nörlund type by using the multivariate fermionic -adic integral on .

2.  -Extensions of Genocchi Numbers and Polynomials of Nörlund Type

In this section, we assume that with . We first consider the -extensions of (1.8) given by the rule

Thus, we obtain the following lemma.

Lemma 2.1. If , then

By (1.14), Thus, we have and we obtain the following theorem.

Theorem 2.2. If , then where stand for the nth Genocchi numbers.

Consider a -extensoin of (1.9) such that and Let . Then, In the special case , the numbers are referred to as -extension of the Genocchi numbers of order . In the sense of the -extension in (1.10), consider the -extension of Genocchi polynomials of Nörlund type given by By (2.8), and . Therefore, we obtain the following theorem.

Theorem 2.3. For , and, , write Then,

The numbers are referred to as the -extension of Genocchi numbers of Nörlund type. For and , introduce the extended higher-order -Genocchi polynomials as follows: Then, Let . Then, we can readily see that Therefore, we obtain the following theorem.

Theorem 2.4. For and , let Then,

Let us now define the extended higher-order Nörlund type -Genocchi polynomials as follows: By (2.16), Let . Then, we have where, . Therefore, we obtain the following theorem.

Theorem 2.5. For , , and , write Then, where, .

For , It can readily be seen that By (2.23), . As is known, It follows from (2.24) that By (2.25), A simple manipulation shows that By (2.27), .

Therefore, we obtain the following proposition.

Proposition 2.6. For , and , the following equations hold. Moreover, .

By (2.21), Hence, For , . It also follows from (2.26) that

The Stirling numbers of the first kind are defined as (see[6, 9]), It can readily be seen that By (2.33) and (2.34), Formulas (2.22) and (2.35) imply the following assertion.

Proposition 2.7. For and ,

The generalized Genocchi numbers and polynomials of Nörlund type are defined by and . We can now also define a -extension of (2.37) as follows. For and , write

and . Thus, Another -extension of Nörlund type generalized Genocchi numbers and polynomials is also of interest, namely, and . By (2.40),

3. Further Remarks

For , consider the following polynomials and : Then, Let and let . Then, Consider the following polynomials: A simple calculation of the fermionic -adic invariant integral on show that By (3.5), . It can readily be proved that By (3.6), . Using (2.24), we can also prove that Thus, . For , we have , where is the Kronecker delta.

It is easy to see that . By (3.4), In particular, if , then for .

Recently, Kim has studied -adic fermionic integral on connected with the problems of mathematical physics (see [6, 10, 11]), and our result are closely related to his results. In the future, we will try to study -adic stochastic problems associated with our theorems. For example, -adic -Bernstein polynomials seem to be closely related to our results (see [6, 14, 20]).

References

  1. I. N. Cangul, H. Ozden, and Y. Simsek, “A new approach to q-Genocchi numbers and their interpolation functions,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 12, pp. e793–e799, 2009. View at Publisher · View at Google Scholar
  2. J. Choi, P. J. Anderson, and H. M. Srivastava, “Some q-extensions of the Apostol-Bernoulli and the Apostol-Euler polynomials of order n, and the multiple Hurwitz zeta function,” Applied Mathematics and Computation, vol. 199, no. 2, pp. 723–737, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. L. Comtet, Advanced Combinatorics, D. Reidel, Dordrecht, The Netherlands, 1974.
  4. E. Y. Deeba and D. M. Rodriguez, “Stirling's series and Bernoulli numbers,” The American Mathematical Monthly, vol. 98, no. 5, pp. 423–426, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. M. Cenkci, M. Can, and V. Kurt, “p-adic interpolation functions and kummer-type congruences for q-twisted euler numbers,” Advanced Studies in Contemporary Mathematics, vol. 9, no. 2, pp. 203–216, 2004. View at Google Scholar
  6. T. Kim, S. D. Kim, and D.-W. Park, “On uniform differentiability and q-Mahler expansions,” Advanced Studies in Contemporary Mathematics, vol. 4, no. 1, pp. 35–41, 2001. View at Google Scholar
  7. 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
  8. T. Kim, “The modified q-Euler numbers and polynomials,” Advanced Studies in Contemporary Mathematics, vol. 16, no. 2, pp. 161–170, 2008. View at Google Scholar
  9. T. Kim, “Note on the Euler q-zeta functions,” Journal of Number Theory, vol. 129, no. 7, pp. 1798–1804, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  10. T. Kim, “q-Volkenborn integration,” Russian Journal of Mathematical Physics, vol. 9, no. 2, pp. 288–299, 2002. View at Google Scholar · View at Zentralblatt MATH
  11. T. Kim, “Euler numbers and polynomials associated with zeta functions,” Abstract and Applied Analysis, Article ID 581582, 11 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. 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. 133–137, 2007. View at Google Scholar
  13. 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
  14. T. Kim, J. Choi, and Y.-H. Kim, “Some identities on the q-Bernstein polynomials, q-Stirling numbers and q-Bernoulli numbers,” Advanced Studies in Contemporary Mathematics, vol. 20, no. 3, pp. 335–341, 2010. View at Google Scholar
  15. T. Kim, “Power series and asymptotic series associated with the q-analog of the two-variable p-adic L-function,” Russian Journal of Mathematical Physics, vol. 12, no. 2, pp. 186–196, 2005. View at Google Scholar
  16. T. Kim, “Non-Archimedean q-integrals associated with multiple Changhee q-Bernoulli polynomials,” Russian Journal of Mathematical Physics, vol. 10, no. 1, pp. 91–98, 2003. View at Google Scholar
  17. T. Kim, “q-Euler numbers and polynomials associated with p-adic q-integrals,” Journal of Nonlinear Mathematical Physics, vol. 14, no. 1, pp. 15–27, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. Q.-M. Luo, “q-extensions for the Apostol-Genocchi polynomials,” General Mathematics, vol. 17, no. 2, pp. 113–125, 2009. View at Google Scholar · View at Zentralblatt MATH
  19. Q.-M. Luo, “Fourier expansions and integral representations for Genocchi polynomials,” Journal of Integer Sequences, vol. 12, no. 1, article 09.1.4, 2009. View at Google Scholar
  20. Q.-M. Luo, “Some results for the q-Bernoulli and q-Euler polynomials,” Journal of Mathematical Analysis and Applications, vol. 363, no. 1, pp. 7–18, 2010. View at Publisher · View at Google Scholar
  21. H. Ozden, Y. Simsek, S.-H. Rim, and I. N. Cangul, “A note on p-adic q-Euler measure,” Advanced Studies in Contemporary Mathematics, vol. 14, no. 2, pp. 233–239, 2007. View at Google Scholar
  22. K.i Shiratani and S. Yamamoto, “On a p-adic interpolation function for the Euler numbers and its derivatives,” Memoirs of the Faculty of Science, Kyushu University A, vol. 39, no. 1, pp. 113–125, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. Y. Simsek and M. Acikgoz, “A new generating function of (q) Bernstein-type polynomials and their interpolation function,” Abstract and Applied Analysis, Article ID 769095, 12 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. Y. Simsek, “On p-Adic Twisted q-L-functions related to generalized twisted bernoulli numbers,” Russian Journal of Mathematical Physics, vol. 13, no. 3, pp. 340–348, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. Y. Simsek, “Theorems on twisted L-function and twisted Bernoulli numbers,” Advanced Studies in Contemporary Mathematics, vol. 11, no. 2, pp. 205–218, 2005. View at Google Scholar
  26. Y. Simsek, “q-Dedekind type sums related to q-zeta function and basic L-series,” Journal of Mathematical Analysis and Applications, vol. 318, no. 1, pp. 333–351, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. H. J. H. Tuenter, “A symmetry of power sum polynomials and Bernoulli numbers,” The American Mathematical Monthly, vol. 108, no. 3, pp. 258–261, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet