On the linearization problem involving Pochhammer symbols and their \(q\)-analogues
type: Item
instance of: scholarly article
title: On the linearization problem involving Pochhammer symbols and their \(q\)-analogues
zbMATH Open document ID: 0934.33026
DOI: 10.1016/S0377-0427(99)00087-4
published in: Journal of Computational and Applied Mathematics
publication date: 2000-04-10T00:00:00Z
review text: Given three sequences \((P_n(x))\), \((Q_n(x))\), \((R_n(x))\) of polynomials, the \(n\)th polynomial in each sequence being of degree \(n\), the linearization problem is the problem of finding the coefficients \(L_{mjn}\) in the expansion \(Q_m(x)R_j(x)=\sum _{n=0} ^{m+j}L_{mjn}P_n(x)\). Under the assumption that the sequences \((Q_n(x))\) and \((R_n(x))\) satisfy second-order \(q\)-difference equations and that the polynomials \((P_n(x))\) are orthogonal and satisfy a fourth-order \(q\)-difference equation, an algorithm is presented which finds a recurrence of the form \(\sum _{k} ^{}c_k(i,j,n)L_{m,j,n+k}=0\) for the coefficients that have to be found. It is shown, in some examples in which the polynomials are shifted \(q\)-factorials and \(q\)-Charlier polynomials, how this recurrence can be used to find closed forms for the coefficients.
Mathematics Subject Classification ID: 33D45
Mathematics Subject Classification ID: 33D60
Mathematics Subject Classification ID: 05A30
zbMATH DE Number: 1326530
zbMATH Keywords: linearization problem
zbMATH Keywords: \(q\)-Pochhammer symbol
zbMATH Keywords: shifted \(q\)-factorial
zbMATH Keywords: \(q\)-Charlier polynomial
Wikidata QID: Q63543376
author: Renato Álvarez-Nodarse
author: Niurka Rodriguez Quintero
author: A. Ronveaux
reviewed by: C. Krattenthaler
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1326530