Decomposition of Laguerre polynomials with respect to the cyclic group of order \(n\)
type: Item
instance of: scholarly article
title: Decomposition of Laguerre polynomials with respect to the cyclic group of order \(n\)
zbMATH Open document ID: 0930.33003
DOI: 10.1016/S0377-0427(98)00145-9
author: Youssèf Ben Cheikh
published in: Journal of Computational and Applied Mathematics
publication date: 2000-02-16T00:00:00Z
review text: Here, the author considers a generalized case of Laguerre polynomial \(L^{(\alpha)}_m\), \[ L^{(\alpha,n,k)}_m(z)= {1\over n} \sum^{n- 1}_{\ell= 0} \exp\Biggl(-{2i\pi k\ell\over n}\Biggr) L^{(\alpha)}_m\Biggl(z\exp \Biggl({2i\pi\ell\over n}\Biggr)\Biggr),\quad z\in\mathbb{C} \] with two additional parameters \(n\) and \(k\). A close relationship between \(L^{(\alpha, n,k)}_m(z)\) and the Brafman polynomials is established. First, a derivation, from the basic identities and relations for \(L^{(\alpha)}_m\), other analogues for \(L^{(\alpha, n,k)}_m\) that turn out to be two integral representations, an operational representation, some generating functions defined by means of the generalized hyperbolic functions of order \(n\) and the hyper-Bessel functions, some finite sums including multiplication and addition formulas, a non-standard \((2n+1)\)-term recurrence relation and a differential equation of order \(2n\). Then some identities of \(L^{(\alpha)}_m\) are expressed as functions of the polynomials \(L^{(\alpha, n,k)}_m\). Finally, some properties of the first component \(L^{(\alpha, n,0)}_m\) are also given.
reviewed by: Arun Kumar Agarwal
Mathematics Subject Classification ID: 33C45
zbMATH DE Number: 1326356
zbMATH Keywords: Laguerre polynomial
zbMATH Keywords: Brafman polynomials
zbMATH Keywords: hyper-Bessel functions
MaRDI profile type: Publication
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full work available at URL: s0377-0427(98)00145-9
OpenAlex ID: W2023223647
description: scientific article; zbMATH DE number 1326356