Research Article | Open Access

# Arithmetic Identities Involving Genocchi and Stirling Numbers

**Academic Editor:**Leonid Berezansky

#### 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 [1]):

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 [2])

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 [2])

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, 3–16]). 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 [17]):
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).

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#### Copyright

Copyright © 2009 Guodong Liu. 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.