Double interpolation of the Euler numbers
type: Item
instance of: scholarly article
title: Double interpolation of the Euler numbers
zbMATH Open document ID: 0821.11009
DOI: 10.1016/0195-6698(95)90088-8
author: Léonce Lesieur
author: Jean-Louis Nicolas
published in: European Journal of Combinatorics
publication date: 1995-09-27T00:00:00Z
review text: The Eulerian numbers, which have well known combinatorial properties, are defined as \[ A(k, n)= \sum_{p=0}^{k-1} (-1)^p \binom{n+1}{p} (k-p)^n \] for \(1\le k\le n\), \(k\), \(n\), integers. The paper gives two methods for interpolating \(A\) by a function of two real variables \(A(x, y)\) (with \(y>-1\)), such that the following properties are retained: \[ A(x, y+1)= (y+2-x) A(x-1, y)+ xA(x, y), \qquad A(x,y)= A(y+1-x,y). \] The first formula for \(A\) is an explicit piecewise polynomial, the second formula is an integral expression, and the main result of the paper is that they give the same function. Some further properties of the function \(A\) are obtained in the paper, among which the computation of the integral \(\int_{-\infty} ^{+\infty} A(x,y) \,dx= \Gamma (y+1)\) which generalizes Worpitzky's formula \(\sum_{k=1}^ n A(k, n)= n!\).
Mathematics Subject Classification ID: 11B68
Mathematics Subject Classification ID: 11B75
zbMATH DE Number: 731894
zbMATH Keywords: Eulerian numbers
zbMATH Keywords: two methods for interpolation
reviewed by: Philippe Biane
MaRDI profile type: Publication
full work available at URL: 0195-6698(95)90088-8
OpenAlex ID: W1994499186
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description: scientific article; zbMATH DE number 731894