Arithmetic Identities Involving Genocchi and Stirling Numbers

Guodong Liu Department of Mathematics, Huizhou University, Huizhou, Guangdong 516015, China Correspondence should be addressed to Guodong Liu, gdliu@pub.huizhou.gd.cn Received 18 June 2009; Accepted 12 August 2009 Recommended by Leonid Berezansky 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. Copyright q 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.


Introduction
The Genocchi numbers G n and the Bernoulli numbers B n n ∈ N 0 {0, 1, 2, . ..} are defined by the following generating functions see 1 : respectively.By 1.1 and 1.2 , we have with N being the set of positive integers.
The Genocchi numbers G n satisfy the recurrence relation or by the generating function It follows from 1.5 or 1.6 that Stirling numbers of the second kind S n, k can be defined by see 2 or by the generating function It follows from 1.8 or 1.9 that 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.

Definition and Lemma
Definition 2.1.For a real or complex parameter x, we have the generalized Genocchi numbers G x n , which are defined by By 1.1 and 2.1 , we have nG Remark 2.2.For an integer x, the higher-order Euler numbers E x 2n are defined by the following generating functions see 17 :

2.3
Then we have where n/2 denotes the greatest integer not exceeding n/2. where 2 n−j S n, j s j, k .

2.6
Proof.By 2.1 , 1.5 , and 1.9 we have which readily yields

2.8
This completes the proof of Lemma 2.3 .
Remark 2.4.From 1.7 , 1.10 , and Lemma 2.3 we know that G x n is a polynomial of x with integral coefficients.For example, setting n 1, 2, 3, 4 in Lemma 2.3, we get

2.10
Therefore, if q ∈ N is odd, then by 2.10 we get where k ∈ N.

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 log 2/ e 2t 1 is the principal branch of logarithm of 2/ e 2t 1 .Thus, by 3.1 and 3.2 , we have

3.3
Now note that whence by integrating from 0 to t, we deduce that  2 n / n 1 G n 1 , we immediately obtain Theorem 1.6.This completes the proof of Theorem 1.6.

6
By 3.6 and 2.6 , we may immediately obtain Theorem 1.1.This completes the proof of Theorem 1.1.