Applications of an explicit formula for the generalized Euler numbers
type: Item
instance of: scholarly article
title: Applications of an explicit formula for the generalized Euler numbers
zbMATH Open document ID: 1145.11020
author: Guodong Liu
author: Wenpeng Zhang
publication date: 2008-05-26T00:00:00Z
review text: For real number \(x\) and positive integers \(n,k\), let \(E^{(x)}_{2n}\), \(s(n,k)\), \(T(n,k)\) denote the generalized Euler numbers, Stirling numbers and central factorial numbers, respectively. In the present paper under review, the authors prove an explicit formula for \(E^{(x)}_{2n}\) as follows: \[ E^{(x)}_{2n}=\sum_{i=1}^n\rho(n,i)x^i, \] here \[ \rho(n,k)=(-1)^k\sum_{j=k}^n\frac{(2j)!}{2^jj!}s(j,k)T(n,j). \] By using this formula, they also obtain some interesting identities and congruences involving the higher-order Euler numbers, Stirling numbers, the central factorial numbers and values of the Riemann zeta function.
reviewed by: Zhefeng Xu
Mathematics Subject Classification ID: 11B68
Mathematics Subject Classification ID: 11B73
Mathematics Subject Classification ID: 11M38
Mathematics Subject Classification ID: 11B83
Mathematics Subject Classification ID: 05A10
zbMATH DE Number: 5278768
zbMATH Keywords: generalized Euler numbers
zbMATH Keywords: Stirling numbers
zbMATH Keywords: central factorial numbers
zbMATH Keywords: congruences
MaRDI profile type: Publication
full work available at URL: s10114-007-1013-x
OpenAlex ID: W2081204777
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DOI: 10.1007/S10114-007-1013-X
published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 5278768