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, 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 [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).