Research Article | Open Access
Arithmetic Identities Involving Genocchi and Stirling Numbers
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.
The Genocchi numbers and the Bernoulli numbers () are defined by the following generating functions (see ):
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 )
or by the generating function
with , , , or .
Stirling numbers of the second kind can be defined by (see )
or by the generating function
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
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
Remark 2.2. For an integer , the higher-order Euler numbers are defined by the following generating functions (see ): Then we have where denotes the greatest integer not exceeding .
Lemma 2.3. Let , then 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
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.
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 © 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.