The contiguous function relations for the basic hypergeometric series
type: Item
instance of: scholarly article
title: The contiguous function relations for the basic hypergeometric series
zbMATH Open document ID: 0703.33012
DOI: 10.1016/0022-247X(90)90292-N
author: Rene F. Swarttouw
published in: Journal of Mathematical Analysis and Applications
publication date: 1990-01-01T00:00:00Z
review text: The contiguous function relations for the basic hypergeometric function \({}_ r\phi_ s(z)\) comprise (i) \(r+s-1\) relations linking \({}_ r\phi_ s(z)\) and two contiguous functions, and (ii) \(r+1\) relations linking \({}_ r\phi_ s(z)\) and \(s+1\) contiguous functions. The variable z appears explicitly only in the latter ones, and in fact linearly. The relations are analogues of Rainville's contiguous function relations for \({}_ rF_ s(z)\), to which they reduce to the limit \(q\to 1\). The results are established by suitable series manipulations. An example is, in the customary notation, \[ (a_ 1-b_ kq^{-1})\phi =(1-b_ kq^{-1})a_ 1\phi (b_ k-)-(1-a_ 1)b_ kq^{-1}\phi (a_ 1+),\quad k\in \{1,...,s\}. \] As an application, q-Laguerre polynomials are considered.
Mathematics Subject Classification ID: 33D60
Mathematics Subject Classification ID: 33D20
zbMATH DE Number: 4152839
zbMATH Keywords: Rainville contiguous function relations
zbMATH Keywords: basic hypergeometric function
reviewed by: Per W. Karlsson
MaRDI profile type: Publication
cites work: Q4159122
cites work: Über Orthogonalpolynome, die q‐Differenzengleichungen genügen
cites work: The q-analogue of the Laguerre polynomials
cites work: The contiguous function relations for _{𝑝}𝐹_{𝑞} with application to Batemean’s 𝐽_{𝑛}^{𝑢,𝜈} and Rice’s 𝐻_{𝑛}(𝜁,𝑝,𝜈)
cites work: Generating Functions for Bessel and Related Polynomials
description: scientific article; zbMATH DE number 4152839