Pfaff's method. II: Diverse applications
type: Item
instance of: scholarly article
title: Pfaff's method. II: Diverse applications
zbMATH Open document ID: 0862.33003
DOI: 10.1016/0377-0427(95)00258-8
author: George E. Andrews
published in: Journal of Computational and Applied Mathematics
publication date: 1997-05-26T00:00:00Z
review text: The paper discusses Pfaff's proof of the Saalschütz summation, which actually preceded Saalschütz' work by hundred years. Set \(S_n(a,b,c) ={}_3F_2{-n,a,b;1 \choose c,1+a +b-c-n}\), and \(\sigma_n (a,b,c) = {(c-a)_n(c-b)_n \over (c)_n (c-a-b)_n}\). The Saalschütz summation states \(S_n(a,b,c) = \sigma_n (a,b,c)\), and Pfaff proved it in the simplest possible way: showed that \(S_n(a,b,c) - S_{n-1} (a,b,c)\) and \(\sigma_n (a,b,c)- \sigma_{n-1} (a,b,c)\) admit the same recurrence. This Pfaffian approach is shown to be effective for Bailey's, Dougall's, Lakin's and Kummer's summation identities. It is noted that the Pfaffian approach seems most effective for balanced and well-poised hypergeometric series. It is often the case that the Pfaffian approach has to prove a cluster of related identities, and not just one of them.
Mathematics Subject Classification ID: 33C20
Mathematics Subject Classification ID: 05A19
zbMATH DE Number: 908382
zbMATH Keywords: Dougall's theorem
zbMATH Keywords: Bailey's theorem
zbMATH Keywords: Saalschutz summation
reviewed by: László A. Székely
MaRDI profile type: Publication
full work available at URL: 0377-0427(95)00258-8
OpenAlex ID: W2088444795
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description: scientific article; zbMATH DE number 908382