Explicit formulas and recurrence relations for higher order Eulerian polynomials
type: Item
instance of: scholarly article
title: Explicit formulas and recurrence relations for higher order Eulerian polynomials
zbMATH Open document ID: 1377.11030
author: Feng Qi
author: Bai-Ni Guo
published in: Indagationes Mathematicae. New Series
publication date: 2017-08-01T00:00:00Z
review text: The higher order Eulerian polynomials \(A_n^{(\alpha)}(t)\) are defined by the exponential generating function \[ \left(\frac{1-t}{e^{x(t-1)}-t}\right)^\alpha=\sum_{n=0}^\infty A_n^{(\alpha)}(t)\frac{x^n}{n!}. \] In this paper the authors prove a simple representation for these numbers in terms of the Euler gamma function and Stirling numbers, find a nonlinear ordinary differential equation for the above generating function (with respect to the variable \(x\)), and find some recurrences for \(A_n^{(\alpha)}(t)\). Among other things, it is proven that the well known relation \[ A_n(t)=\sum_{k=0}^nk!S(n,k)(t-1)^{n-k} \] generalizes to \[ A_n^{(\alpha)}(t)=\frac{1}{\Gamma(\alpha)}\sum_{k=0}^n\Gamma(\alpha+k)S(n,k)(t-1)^{n-k}. \]
Mathematics Subject Classification ID: 11B68
Mathematics Subject Classification ID: 11B37
zbMATH DE Number: 6755373
zbMATH Keywords: Eulerian polynomial
zbMATH Keywords: higher order Eulerian polynomial
zbMATH Keywords: Stirling numbers of the second kind
reviewed by: István Mező
MaRDI profile type: Publication
full work available at URL: j.indag.2017.06.010
OpenAlex ID: W2724722377
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DOI: 10.1016/J.INDAG.2017.06.010
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description: scientific article; zbMATH DE number 6755373