Arithmetic Identities Involving Genocchi and Stirling Numbers

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.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Recommended by Leonid Berezansky

Department of Mathematics, Huizhou University, Huizhou, Guangdong 516015, China

Received 18 June 2009; Accepted 12 August 2009

1. Introduction

The Genocchi numbers _Gn and the Bernoulli numbers _Bn (n∈_...0 ={0,1,2,...} ) are defined by the following generating functions (see [1]):

[figure omitted; refer to PDF] [figure omitted; refer to PDF] respectively. By (1.1) and (1.2), we have

[figure omitted; refer to PDF] with ... being the set of positive integers.

The Genocchi numbers _Gn satisfy the recurrence relation

[figure omitted; refer to PDF] so we find _G2 =-1, _G4 =1, _G6 =-3, _G8 =17, _G10 =-155, _G12 =2073, _G14 =-38227,....

The Stirling numbers of the first kind s(n,k) can be defined by means of (see [2])

[figure omitted; refer to PDF] or by the generating function

[figure omitted; refer to PDF]

It follows from (1.5) or (1.6) that

[figure omitted; refer to PDF] with s(n,0)=0 (n>0) , s(n,n)=1 (n≥0) , s(n,1)=^(-1)n-1 (n-1)! (n>0) , s(n,k)=0 (k>n or k<0) .

Stirling numbers of the second kind S(n,k) can be defined by (see [2])

[figure omitted; refer to PDF] or by the generating function

[figure omitted; refer to PDF] It follows from (1.8) or (1.9) that

[figure omitted; refer to PDF] with S(n,0)=0 (n>0) , S(n,n)=1 (n≥0) , S(n,1)=1 (n>0) , S(n,k)=0 (k>n or k<0) .

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 n≥k (n,k∈...) , then [figure omitted; refer to PDF]

Remark 1.2.

Setting k=1 in (1.11), and noting that s(j,1)=^(-1)j-1 (j-1)! , we obtain [figure omitted; refer to PDF]

Remark 1.3.

By (1.11) and (1.3), we have [figure omitted; refer to PDF]

Theorem 1.4.

Let n,k∈... , then [figure omitted; refer to PDF]

Remark 1.5.

Setting k=1,2,3,4 in (1.14), we get [figure omitted; refer to PDF]

Theorem 1.6.

Let n∈... , m∈_...0 , then [figure omitted; refer to PDF]

Remark 1.7.

Setting m=p-1 in (1.16), we have [figure omitted; refer to PDF] where p is any odd prime.

2. Definition and Lemma

Definition 2.1.

For a real or complex parameter x , we have the generalized Genocchi numbers ^Gn(x) , which are defined by [figure omitted; refer to PDF] By (1.1) and (2.1), we have [figure omitted; refer to PDF]

Remark 2.2.

For an integer x , the higher-order Euler numbers ^E2n(x) are defined by the following generating functions (see [17]): [figure omitted; refer to PDF] Then we have [figure omitted; refer to PDF] where [n/2] denotes the greatest integer not exceeding n/2 .

Lemma 2.3.

Let n≥k (n,k∈...) , then [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Proof.

By (2.1), (1.5), and (1.9) we have [figure omitted; refer to PDF] which readily yields [figure omitted; refer to PDF] This completes the proof of Lemma 2.3.

Remark 2.4.

From (1.7), (1.10), and Lemma 2.3 we know that ^Gn(x) is a polynomial of x with integral coefficients. For example, setting n=1,2,3,4 in Lemma 2.3, we get [figure omitted; refer to PDF]

Remark 2.5.

Let n,m∈... , then by (2.5), we have [figure omitted; refer to PDF] Therefore, if q∈... is odd, then by (2.10) we get [figure omitted; refer to PDF] where k∈... .

3. Proof of the Theorems

Proof of Theorem 1.1.

By applying Lemma 2.3, we have [figure omitted; refer to PDF] On the other hand, it follows from (2.1) that [figure omitted; refer to PDF] where log (2/(^e2t +1)) is the principal branch of logarithm of 2/^(e2t +1).

Thus, by (3.1) and (3.2), we have [figure omitted; refer to PDF]

Now note that [figure omitted; refer to PDF] whence by integrating from 0 to t , we deduce that [figure omitted; refer to PDF] Since _G2n+1 =0 (n∈... ). Substituting (3.5) in (3.3) we get [figure omitted; refer to PDF]

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 [figure omitted; refer to PDF] we have [figure omitted; refer to PDF]

By (3.8), (1.7), and note that ^Gn(1) =²ⁿ /(n+1)_Gn+1 , we obtain [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] Therefore [figure omitted; refer to PDF] Taking k=1 in (3.11) and note that ^Gn(1) =²ⁿ /(n+1)_Gn+1 , 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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