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Pfaff’s method. II: Diverse applications. (English) Zbl 0862.33003
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.

MSC:
33C20 Generalized hypergeometric series, \({}_pF_q\)
05A19 Combinatorial identities, bijective combinatorics
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