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Mathematical Problems in Engineering
Volume 2010 (2010), Article ID 316870, 8 pages
http://dx.doi.org/10.1155/2010/316870
Research Article

On Multiple Generalized -Genocchi Polynomials and Their Applications

Department of Mathematics and Computer Science, Konkuk University, Chungju, Chungcheongbuk-do 380-701, Republic of Korea

Received 26 September 2010; Accepted 3 December 2010

Academic Editor: Ben T. Nohara

Copyright © 2010 Lee-Chae Jang. 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 define the multiple generalized -Genocchi polynomials. By using fermionic -adic invariant integrals, we derive some identities on these generalized -Genocchi polynomials, for example, fermionic -adic integral representation, Witt's type formula, explicit formula, multiplication formula, and recurrence formula for these -Genocchi polynomials.

1. Introduction

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

The -basic natural numbers are defined by for , and the binomial coefficient is defined as The binomial formulas are well known that (see, [1, 2]). When one talks of -extension, is variously considered as an indeterminate, a complex number , or a -adic number . If , one normally assumes that . If , one normally assumes that . We use the notation see [113] for all . Note that for in presented -adic case.

Let be denoted by the set of uniformly differentiable function on . For , an invariant -adic -integral on is defined as Thus, we have the following integral relation: where , and the fermionic -adic invariant integral relation:

Now, we recall that the definitions of -Euler polynomials and -Genocchi polynomials are defined as with , respectively. In the special case , , and are called -Euler numbers and -Genocchi numbers (see [2, 9]).

In [13], Bayard and Simsek have studied multiple generalized Bernoulli polynomials as follows: where are strictly positive real numbers.

The purpose of this paper is to define another construction of multiple generalized -Genocchi polynomials and numbers, which are different from multiple generalized Bernoulli polynomials and numbers in [13]. By using fermionic -adic invariant integrals, we derive some identities on these generalized -Genocchi polynomials, for example, fermionic -adic integral representation, Witt's type formula, explicit formulas, multiplication formula, and recurrence formula for these -Genocchi polynomials.

2. Multiple Generalized -Genocchi Polynomials and Numbers

Let and be strictly positive real numbers. The multiple generalized -Genocchi polynomials are defined as where . The values of at are called the multiple generalized -Genocchi numbers: when , , and (), the polynomials or numbers are called the ordinary Genocchi polynomials or numbers.

It is known that

In fact, let us take , and we apply the above difference integral formula (1.8) for , then we obtain By (2.3), we easily see that

By (2.4) and (2.5), we obtain the following fermionic -adic integral representation formula for these numbers.

Theorem 2.1 (-adic integral representation). Let and be strictly positive real numbers. Then one has a fermionic -adic invariant integral representation for the multiple generalized -Genocchi polynomials as follows: for and

We remark that if we set and , then we have the following equation: The generalized -Genocchi polynomials are given by By comparing the coefficients on both sides in (2.9), we obtain the following identity on the generalized -Genocchi polynomials Similarly, from (2.4), we can obtain the following Witt's type formula for the multiple generalized -Genocchi polynomials.

Theorem 2.2 (Witt's type formula). Let and be strictly positive real numbers. Then one has

From (2.4), we can directly calculate the following: From (2.12), we get the following explicit formula.

Theorem 2.3 (explicit formula). Let and be strictly positive real numbers. Then one has

Next we discuss the multiplication formula for the multiple generalized -Genocchi polynomials as follows: Thus, we obtain the following multiplication formula for the multiple generalized -Genocchi polynomials.

Theorem 2.4 (multiplication formula). Let and be strictly positive real numbers. For any , one has

Corollary 2.5. (1) If one sets and , then one obtains Raabe type formula for multiple Genocchi polynomials as follows: where .
(2) If one sets and , then one obtains Carlitz's multiplication formula for the multiple generalized Genocchi polynomials as follows: where .

Finally, we discuss the recurrence formula for the multiple generalized -Genocchi polynomials as follows. Let and be strictly positive real numbers. For any , we can directly derive the following equation: By comparing the coefficients on both sides in (2.18), we obtain the recurrence formula for the multiple generalized -Genocchi polynomials.

Theorem 2.6 (recurrence formula). Let and be strictly positive real numbers. For any , one has

Acknowledgment

This paper was supported by Konkuk University in 2011.

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, L.-C. Jang, and H. K. Pak, “A note on q-Euler and Genocchi numbers,” Proceedings of the Japan Academy, Series A, vol. 77, no. 8, pp. 139–141, 2001. View at Publisher · View at Google Scholar
  3. L. Carlitz, “The multiplication formulas for the Bernoulli and Euler polynomials,” Mathematics Magazine, vol. 27, pp. 59–64, 1953. View at Google Scholar · View at Zentralblatt MATH
  4. L. Jang and T. Kim, “q-Genocchi numbers and polynomials associated with fermionic p-adic invariant integrals on p,” Abstract and Applied Analysis, vol. 2008, Article ID 232187, 8 pages, 2008. View at Google Scholar
  5. T. Kim, “New approach to q-Euler polynomials of higher order,” Russian Journal of Mathematical Physics, vol. 17, no. 2, pp. 218–225, 2010. View at Publisher · View at Google Scholar
  6. T. Kim, “New approach to q-Euler, Genocchi numbers and their interpolation functions,” Advanced Studies in Contemporary Mathematics (Kyungshang). Memoirs of the Jangjeon Mathematical Society, vol. 18, no. 2, pp. 105–112, 2009. View at Google Scholar
  7. T. Kim, “Barnes-type multiple q-zeta functions and q-Euler polynomials,” Journal of Physics A, vol. 43, no. 25, Article ID 255201, 11 pages, 2010. View at Publisher · View at Google Scholar
  8. 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
  9. 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
  10. T. Kim, “On p-adic q-l-functions and sums of powers,” Journal of Mathematical Analysis and Applications, vol. 329, no. 2, pp. 1472–1481, 2007. View at Publisher · View at Google Scholar
  11. T. Kim, “On the symmetries of the q-Bernoulli polynomials,” Abstract and Applied Analysis, vol. 2008, Article ID 914367, 7 pages, 2008. View at Google Scholar
  12. 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
  13. A. Bayard and Y. Simsek, “On multiple generalized Bernoulli numbers and polynomials and their Barnes-Lerch zeta functions,” preprint.