Andrews, George E. Pfaff’s method. II: Diverse applications. (English) Zbl 0862.33003 J. Comput. Appl. Math. 68, No. 1-2, 15-23 (1996). 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. Reviewer: L.A.Székely (Columbia/South Carolina Cited in 3 ReviewsCited in 9 Documents MSC: 33C20 Generalized hypergeometric series, \({}_pF_q\) 05A19 Combinatorial identities, bijective combinatorics Keywords:Dougall’s theorem; Bailey’s theorem; Saalschutz summation PDF BibTeX XML Cite \textit{G. E. Andrews}, J. Comput. Appl. Math. 68, No. 1--2, 15--23 (1996; Zbl 0862.33003) Full Text: DOI OpenURL Digital Library of Mathematical Functions: §17.7(iii) Other ϕ s r Functions ‣ §17.7 Special Cases of Higher ϕ s r Functions ‣ Properties ‣ Chapter 17 q -Hypergeometric and Related Functions References: [1] Andrews, G.E., On the q-analog of Kummer’s theorem and applications, Duke math. J., 40, 525-528, (1973) · Zbl 0266.33003 [2] Andrews, G.E., Connection coefficient problems and partitions, (), 1-24 · Zbl 0186.30203 [3] Andrews, G.E., Plane partitions III: the weak Macdonald conjecture, Invent. math., 53, 193-225, (1979) · Zbl 0421.10011 [4] G.E. Andrews, Pfaff’s method I: the Mills-Robbins-Rumsey determinant, Discrete Math., to appear. · Zbl 1069.15009 [5] Andrews, G.E.; Burge, W.H., Determinant identities, Pacific J. math., 158, 1-14, (1993) · Zbl 0793.15001 [6] G.E. Andrews and D.W. Stanton, Determinants in plane partitions enumeration, to appear. · Zbl 0908.05007 [7] Askey, R., Variants of Clausen’s formula for the square of special _{2}F1, (), 1-12, Bombay · Zbl 0756.33002 [8] Bailey, W.N., Some identities involving generalized hypergeometric series, (), 503-516, (2) · JFM 55.0219.05 [9] Bailey, W.N., Generalized hypergeometric series, (1935), Cambridge Univ. Press London and New York, (Reprinted: Hafner, New York, 1964) · Zbl 0011.02303 [10] Burchnall, J.L.; Lakin, A., The theorems of saalschutz and dougall, Quart. J. math. Oxford, 1, 2, 161-164, (1950) · Zbl 0040.03401 [11] Daum, J.A., The basic analog of Kummer’s theorem, Bull. amer. math. soc., 48, 711-713, (1942) · Zbl 0060.19808 [12] Dougall, J., On Vandermonde’s theorem and some more general expansions, (), 114-132 · JFM 38.0313.01 [13] Gessel, I.; Stanton, D., Strange evaluations of hypergeometric series, SIAM J. math. anal., 13, 295-308, (1982) · Zbl 0486.33003 [14] Jackson, F.H., Transformations of q-series, Messenger math., 39, 145-153, (1910) [15] Jackson, F.H., Summation of q-hypergeometric series, Messenger math., 50, 101-112, (1921) [16] Lakin, A., A hypergeometric identity related to Dougall’s theorem, J. London math. soc., 27, 229-234, (1952) · Zbl 0046.07401 [17] Mills, W.H.; Robbins, D.P.; Rumsey, H., Enumeration of a symmetry class of plane partitions, Discrete math., 67, 43-55, (1987) · Zbl 0656.05006 [18] Pfaff, J.F., Observationes analyticae ad L. Euler institutiones calculi integralis, vol. IV, supplem. II et IV, historia de 1793, Nova acta acad. sci. petropolitanae, 11, 38-57, (1797) [19] Saalschultz, L., Eine summationsformel, Z. math. phys., 35, 186-188, (1890) · JFM 22.0262.03 [20] Wilf, H.S.; Zeilberger, D., Rational functions certify combinatorial identities, J. amer. math. soc., 3, 147-158, (1990) · Zbl 0695.05004 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.