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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.

33C20 Generalized hypergeometric series, \({}_pF_q\)
05A19 Combinatorial identities, bijective combinatorics
Full Text: DOI
[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 _2F1, (), 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
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