Abstract
In this paper, we discuss the properties of a class of Genocchi numbers by generating functions and Riordan arrays, we establish some identities involving Genocchi numbers, the Stirling numbers, the generalized Stirling numbers, the higher order Bernoulli numbers and Cauchy numbers. Further, we get asymptotic value of some sums relating the Genocchi numbers.
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1 Introduction and preliminaries
The Genocchi numbers are defined by
The relationship between the Genocchi numbers and the Bernoulli numbers and the Euler polynomials is
They are known as follows [1].
Genocchi numbers and Genocchi polynomials are prolific in the mathematical literature, and many results on Genocchi numbers and Genocchi polynomials identities may be seen in the papers [2–4]. In this paper, we will mainly study the products of Genocchi numbers with the following forms:
where n is a nonnegative positive integer , and for , for .
It is clear that
Then we have
where .
The paper is organized as follows. In Section 2, we obtain some properties for by means of the method of generating function. In Section 3, we establish some identities involving , the Stirling numbers, the generalized Stirling numbers, the higher order Bernoulli numbers and the Cauchy numbers. Finally, in Section 4, we give the asymptotic expansion of some sums involving .
For convenience, we recall some definitions and notations involved in the paper. Stirling number of the first kind is denoted by , let , , , , stand for the higher order Bernoulli numbers, products of Cauchy numbers, Bell polynomials and the potential polynomials. The corresponding generating functions are
In this paper, we let denote the coefficient of in , where .
An exponential Riordan array is a pair of formal power series. Where , .
It defines an infinite, lower triangular array according to the rule
Hence we write .
The most important property of Riordan array is: If is an exponential Riordan array, and let be the exponential generating function of the sequence , then we have
where we use the notation as a linearization of the more common one to denote substitution .
A singularity of at is called algebraic if can be written as the sum of a function analytic in a neighborhood of w and a finite number of terms of the form
where is analytic near w, , and (see [5]).
Lemma 1.1 (See [5])
Suppose that is analytic for , , and has only algebraic singularities of on . Let a be the minimum of for the terms of the form at the singularity of on , and let , , and be the w, α, and for those terms of form (1.11), for which . Then, as ,
2 Properties of products of Genocchi numbers
In this section, we obtain some properties for by means of the method of generating functions and the Euler transformation.
Theorem 2.1 Let be any integers, then
where .
Proof By (1.1), we have
and the comparison of the coefficients of in the first and the last formulas completes the proof. □
Theorem 2.2 Let , be any integers, then
Proof Deriving both sides of (1.1) with respect to t, we get
Multiplying both sides by t, we have
and the comparison of the coefficients of in the first and the last formulas completes the proof. □
Theorem 2.3 Let be any integers, then
Proof By (1.1) and (1.7), we get
and the comparison of the coefficients of in the first and the last formulas completes the proof. □
Theorem 2.4 Let be any integers, then
Proof By (1.1), (1.8), (1.9) and Theorem 2.3, we have
Then
which completes the proof. □
Theorem 2.5 Let be any integers, then
Proof By (1.2) and (1.5), we have
and the comparison of the coefficients of in the first and the last formulas completes the proof. □
Theorem 2.6 Let be any integers, then
Proof Let , and by (1.10), we have
which completes the proof. □
3 Identities involving , , , and
In this section, by using the Riordan and generating functions method, we explore relationships between these numbers, the , , , and .
Theorem 3.1 Let be any integers, then
where are the Stirling numbers of the first kind, .
Proof By (1.1), (1.5), we have
which yields Theorem 3.1. □
Theorem 3.2 Let be any integers, then
Proof Since
then by .
From (1.10), we have
which completes the proof. □
By the proof of Theorem 3.2, we can get the following corollary.
Corollary 3.3 Let , be any integer, then
Theorem 3.4 Let , be any integer, let be a real number, then
Proof By (1.3) and (1.10), we have
□
Theorem 3.5 Let be any integers, let be a real number, then
The proof of Theorem 3.5 is similar to that of Theorem 3.4, and is omitted here.
Theorem 3.6 Let be any integers, let be a real number, then
Proof For .
By (1.2), (1.3) and (1.10), we have
which completes the proof. □
Theorem 3.7 Let be any integers, let be a real number, then
Proof Since , then by (1.2) and (1.10), we have
which completes the proof. □
Theorem 3.8 Let be any integers, and let be a real number, then
Proof Since , then by (1.3), (1.8) and (1.10), we have
□
Theorem 3.9 Let be any integer, and let be a real number, then
Proof Since , then by (1.2) and (1.10), we have
which completes the proof. □
4 Asymptotics
In this section, we give the asymptotic expansion of certain sums involving .
Theorem 4.1 Let be a real number, and suppose that , then
Proof From the proof of Theorem 3.6, we know
In Lemma 1.1, let , . Obviously, is analytic for , and it has only algebraic singularity on , namely at , so by Lemma 1.1 and Stirling’s formula , we have
which completes the proof. □
Theorem 4.2 Let be any integer, let be a real number, and suppose that , then
Proof By Lemma 1.1 and Stirling’s formula, we have
which completes the proof. □
Theorem 4.3 Let be a real number, as , and suppose that
Proof By Lemma 1.1 and Stirling’s formula, we have
which completes the proof. □
References
Comtet L: Advanced Combinatorics. Reidel, Dordrecht; 1974.
Dumont D: Sur une conjecture de Gandhi concernant les nombres de Genocchi. Discrete Math. 1972, 1: 321–327. 10.1016/0012-365X(72)90039-8
Dumont D: Interpretations combinatoires des nombres de Genocchi. Duke Math. J. 1974, 41: 305–318. 10.1215/S0012-7094-74-04134-9
Dumont D, Foata D: Une propriété de symetrie des nombres de Genocchi. Bull. Soc. Math. Fr. 1976, 104: 433–451.
Odlyzko A: Asymptotic enumeration methods. In Handbook of Combinatorics. Edited by: Lovasz L, Graham R, Grötschel M. Elsevier, Amsterdam; 1995.
Acknowledgements
The author thanks the anonymous referees for their careful reading and valuable comments. The research is supported by the Natural Science Foundation of China under Grant 11061020 and the Natural Science Foundation of Inner Mongolia under Grant 2012MS0118.
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Wuyungaowa Identities on products of Genocchi numbers. J Inequal Appl 2013, 422 (2013). https://doi.org/10.1186/1029-242X-2013-422
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DOI: https://doi.org/10.1186/1029-242X-2013-422