Explicit and recurrence formulas for generalized Euler numbers
type: Item
instance of: scholarly article
title: Explicit and recurrence formulas for generalized Euler numbers
zbMATH Open document ID: 0827.11011
author: Pavel G. Todorov
published in: Functiones et Approximatio. Commentarii Mathematici
publication date: 1995-12-11T00:00:00Z
review text: The generalized Euler numbers \(E_{2n}^{(r)}\) for a complex number \(r\) and positive integer \(n\) are defined by \[ \text{sec}^r z=1/ \cos^r z=1+ \sum_{n=1}^\infty E_{2n}^{(r)} z^{2n}/ (2n)! , \qquad |z|< \pi/2. \tag{1} \] For \(r=1\) we get the ordinary Euler numbers \(E_{2n}\). The author proves the following explicit formula for the generalized Euler numbers: \[ E_{2n}^{(r)}= \sum_{k=1}^n \sum_{j=1}^k (-1)^{n-j} {\textstyle {{r+k-1} \choose k} {{2k} \choose {k-j}}} j^{2n}/ 2^{k-1}, \tag{3} \] which generalizes the known formula for the Euler numbers [\textit{L. Comtet}, Advanced combinatorics (1974; Zbl 0283.05001)]. Furthermore the author gives the recursive formula: \[ E_{2n}^{(r)}= \sum_{m=0}^{n-1} (-1)^{n-1-m} \left[ {\textstyle {{2n-1} \choose {2m}} r+ {{2n-1} \choose {2m-1}}} \right] E_{2m}^{(r)} \tag{8} \] for positive integers \(n\), where \({{2n-1} \choose {-1}} =0\) and \(E_0^{(r)} =1\). We get from the formula (3) that \(E_{2n}^{(r)}\) are polynomials of degree \(n\) and from the formula (8) that the coefficients of these polynomials are integers. Finally it is proved that these coefficients are positive.
Mathematics Subject Classification ID: 11B68
Mathematics Subject Classification ID: 05A15
zbMATH DE Number: 713621
zbMATH Keywords: generalized Euler numbers
zbMATH Keywords: explicit formula
zbMATH Keywords: recursive formula
reviewed by: Ladislav Skula
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 713621