The generalized Touchard polynomials revisited
type: Item
instance of: scholarly article
title: The generalized Touchard polynomials revisited
zbMATH Open document ID: 1301.11028
author: Toufik Mansour
author: Matthias Schork
published in: Applied Mathematics and Computation
publication date: 2014-06-20T00:00:00Z
review text: Starting with the Touchard polynomials, defined by \[ T_n(x)=e^{-x}\left( x \frac{d}{dx}\right) ^ne^x \] for every \(n\in \mathbb N\), several generalizations were recently defined, e.g., for higher order \[ T_n^{(m)}(x)=e^{-x}\left( x^m\frac{d}{dx}\right) ^ne^x \] for positive but also negative integers \(m\). The paper deals extensively with the case \(m=-1\) and among other things suggests to consider generalized Touchard functions of arbitrary real order. As a particular example the case of order \(m=\frac{1}{2}\) is discussed and it is shown that the associated Touchard functions are given by Hermite polynomials. Further generalizations are given (e.g., so called Comtet-Touchard functions associated to powers of an arbitrary derivation are introduced).
Mathematics Subject Classification ID: 11B73
Mathematics Subject Classification ID: 11B83
Mathematics Subject Classification ID: 33C10
zbMATH DE Number: 6305733
zbMATH Keywords: Touchard polynomial
zbMATH Keywords: Stirling number
zbMATH Keywords: Bell number
zbMATH Keywords: generating function
reviewed by: Grigore Călugăreanu
describes a project that uses: OEIS
MaRDI profile type: Publication
full work available at URL: j.amc.2013.04.010
OpenAlex ID: W2026571809
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DOI: 10.1016/J.AMC.2013.04.010
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description: scientific article; zbMATH DE number 6305733