Research Article | Open Access
Guodong Liu, "Arithmetic Identities Involving Genocchi and Stirling Numbers", Discrete Dynamics in Nature and Society, vol. 2009, Article ID 621068, 8 pages, 2009. https://doi.org/10.1155/2009/621068
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.
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.