Research Article | Open Access

Volume 2009 |Article ID 621068 | 8 pages | https://doi.org/10.1155/2009/621068

# Arithmetic Identities Involving Genocchi and Stirling Numbers

Accepted12 Aug 2009
Published27 Oct 2009

#### Abstract

An explicit formula, the generalized Genocchi numbers, was established and some identities and congruences involving the Genocchi numbers, the Bernoulli numbers, and the Stirling numbers were obtained.

#### 1. Introduction

The Genocchi numbers and the Bernoulli numbers () are defined by the following generating functions (see ):

respectively. By (1.1) and (1.2), we have

with being the set of positive integers.

The Genocchi numbers satisfy the recurrence relation

so we find

The Stirling numbers of the first kind can be defined by means of (see )

or by the generating function

It follows from (1.5) or (1.6) that

with , , , or .

Stirling numbers of the second kind can be defined by (see )

or by the generating function

It follows from (1.8) or (1.9) that

with , , , or .

The study of Genocchi numbers and polynomials has received much attention; numerous interesting (and useful) properties for Genocchi numbers can be found in many books (see [1, 316]). The main purpose of this paper is to prove an explicit formula for the generalized Genocchi numbers (cf. Section 2). We also obtain some identities congruences involving the Genocchi numbers, the Bernoulli numbers, and the Stirling numbers. That is, we will prove the following main conclusion.

Theorem 1.1. Let , then

Remark 1.2. Setting in (1.11), and noting that , we obtain

Remark 1.3. By (1.11) and (1.3), we have

Theorem 1.4. Let , then

Remark 1.5. Setting in (1.14), we get

Theorem 1.6. Let , , then

Remark 1.7. Setting in (1.16), we have where is any odd prime.

#### 2. Definition and Lemma

Definition 2.1. For a real or complex parameter , we have the generalized Genocchi numbers , which are defined by By (1.1) and (2.1), we have

Remark 2.2. For an integer , the higher-order Euler numbers are defined by the following generating functions (see ): Then we have where denotes the greatest integer not exceeding .

Lemma 2.3. Let , then where

Proof. By (2.1), (1.5), and (1.9) we have which readily yields This completes the proof of Lemma 2.3.

Remark 2.4. From (1.7), (1.10), and Lemma 2.3 we know that is a polynomial of with integral coefficients. For example, setting in Lemma 2.3, we get

Remark 2.5. Let , then by (2.5), we have Therefore, if is odd, then by (2.10) we get where .

#### 3. Proof of the Theorems

Proof of Theorem 1.1. By applying Lemma 2.3, we have On the other hand, it follows from (2.1) that where is the principal branch of logarithm of
Thus, by (3.1) and (3.2), we have
Now note that whence by integrating from to , we deduce that Since   (). Substituting (3.5) in (3.3) we get
By (3.6) and (2.6), we may immediately obtain Theorem 1.1. This completes the proof of Theorem 1.1.

Proof of Theorem 1.4. By (2.1) and note the identity we have
By (3.8), (1.7), and note that , we obtain Comparing (3.9) and (2.8), we immediately obtain Theorem 1.4. This completes the proof of Theorem 1.4.

Proof of Theorem 1.6. By Lemma 2.3, we have Therefore Taking in (3.11) and note that we immediately obtain Theorem 1.6. This completes the proof of Theorem 1.6.

#### Acknowledgments

The author would like to thank the anonymous referee for valuable suggestions. This work was supported by the Guangdong Provincial Natural Science Foundation (no. 8151601501000002).

1. F. T. Howard, “Applications of a recurrence for the Bernoulli numbers,” Journal of Number Theory, vol. 52, no. 1, pp. 157–172, 1995.
2. J. Riordan, Combinatorial Identities, John Wiley & Sons, New York, NY, USA, 1968. View at: MathSciNet
3. I. N. Cangul, H. Ozden, and Y. Simsek, “A new approach to $q$-Genocchi numbers and their interpolation functions,” Nonlinear Analysis: Theory, Methods & Applications. In press. View at: Publisher Site | Google Scholar
4. M. Cenkci, M. Can, and V. Kurt, “$q$-extensions of Genocchi numbers,” Journal of the Korean Mathematical Society, vol. 43, no. 1, pp. 183–198, 2006.
5. L. Jang and T. Kim, “$q$-Genocchi numbers and polynomials associated with fermionic $p$-adic invariant integrals on ${ℤ}_{p}$,” Abstract and Applied Analysis, vol. 2008, Article ID 232187, 8 pages, 2008. View at: Publisher Site | Google Scholar | MathSciNet
6. T. Kim, “A note on the $q$-Genocchi numbers and polynomials,” Journal of Inequalities and Applications, vol. 2007, Article ID 71452, 8 pages, 2007. View at: Publisher Site | Google Scholar | MathSciNet
7. 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 Site | Google Scholar | MathSciNet
8. 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 Site | Google Scholar | MathSciNet
9. 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 Site | Google Scholar | MathSciNet
10. G. D. Liu and R. X. Li, “Sums of products of Euler-Bernoulli-Genocchi numbers,” Journal of Mathematical Research and Exposition, vol. 22, no. 3, pp. 469–475, 2002 (Chinese).
11. H. Ozden and Y. Simsek, “A new extension of $q$-Euler numbers and polynomials related to their interpolation functions,” Applied Mathematics Letters, vol. 21, no. 9, pp. 934–939, 2008. View at: Google Scholar | MathSciNet
12. H. Ozden, I. N. Cangul, and Y. Simsek, “Multivariate interpolation functions of higher-order $q$-Euler numbers and their applications,” Abstract and Applied Analysis, vol. 2008, Article ID 390857, 16 pages, 2008. View at: Publisher Site | Google Scholar | MathSciNet
13. S.-H. Rim, K. H. Park, and E. J. Moon, “On Genocchi numbers and polynomials,” Abstract and Applied Analysis, vol. 2008, Article ID 898471, 7 pages, 2008. View at: Publisher Site | Google Scholar | MathSciNet
14. C. S. Ryoo, “A numerical computation on the structure of the roots of $q$-extension of Genocchi polynomials,” Applied Mathematics Letters, vol. 21, no. 4, pp. 348–354, 2008. View at: Google Scholar | MathSciNet
15. C. S. Ryoo, “A numerical investigation on the structure of the roots of $q$-Genocchi polynomials,” Journal of Applied Mathematics and Computing, vol. 26, no. 1-2, pp. 325–332, 2008. View at: Publisher Site | Google Scholar | MathSciNet
16. Y. Simsek, I. N. Cangul, V. Kurt, and D. Kim, “$q$-Genocchi numbers and polynomials associated with $q$-Genocchi-type $l$-functions,” Advances in Difference Equations, vol. 2008, Article ID 815750, 12 pages, 2008. View at: Publisher Site | Google Scholar | MathSciNet
17. Y. Simsek, “Complete sums of products of $\left(h,q\right)$-extension of Euler numbers and polynomials,” http://arxiv.org/abs/0707.2849. View at: Google Scholar

#### More related articles

We are committed to sharing findings related to COVID-19 as quickly and safely as possible. Any author submitting a COVID-19 paper should notify us at help@hindawi.com to ensure their research is fast-tracked and made available on a preprint server as soon as possible. We will be providing unlimited waivers of publication charges for accepted articles related to COVID-19. Sign up here as a reviewer to help fast-track new submissions.