Inversion techniques and combinatorial identities: a unified treatment for the 7F6 series identities

• Autores: Chu Wenchang
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 45, Fasc. 1, 1994, págs. 13-44
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • As an extension of a useful inverse pair due to Gould–Hsu (1973), a general pair of reciprocal relations is established. The inversion technique for proving combinatorial identities, originated by Riordan (1968) and Greene & Knuth (1981), is developed systematically to explore the dual relations of Pfaff–Saalschutz and Dougall–Dixon formulae. Most of the known strange hypergeometric evaluations, covered in Bailey (1935), Slater (1966), Gessel & Staton (1982), Gasper (1989) and Gasper & Rahman (1990), and several new mysterious looking formulas are demonstrated almost mechanically.

• Referencias bibliográficas
  • G. E. Andrews, Connection coefficient problems and partitions, Proc. Symposia in Pure Math. 34 (1979), 1–24.
  • R. Askey, Variants of Clausen’s formula for the square of a special ${}\_2F\_1$. In: Number Theory and Related Topics. (pp. 1–12) Bombay:...
  • W. N. Bailey, Some identities involving generalized hypergeometric series, Proc. London Math. Soc. 29 (1929), 503–516.
  • W. N. Bailey, Generalized Hypergeometric Series. London: Cambridge University Press 1935.
  • D. M. Bressoud, The Bailey lattice: an introduction, 57–67. In: Andrews, G. E., et al (eds.) Ramanujan Revisited, Proceedings, Urbana (1987)....
  • W. C. Chu, Inversion techniques and combinatorial identities. Strange evaluations of hypergeometric series. Roma. Preprint 1990.
  • W. C. Chu and L. C. Hsu, Some new applications of Gould–Hsu inversions. J. of Combin., Informat. and Systems Science, 14:1 (1990), 1–4.
  • S. B. Ekhad, A short proof of a terminating “strange” hypergeometric summation formula of Gasper and Rahman. Preprint 1990.
  • G. Gasper, Summation, transformation and expansion formulas for bibasic series. Trans. Amer. Math. Soc. 312:1, (1989), 257–277.
  • G. Gasper and M. Rahman, An indefinite bibasic summation formula and some quadratic, cubic and quartic summations and transformation formulas....
  • G. Gasper and M. Rahman, Basic Hypergeometric Series. (Rota, G. C. (ed.) Encyclopedia of Mathematics and Its Application, vol. 35) Cambridge:...
  • Ira Gessel and D. Stanton, Strange evaluations of hypergeometric series. SIAM J. Math. Anal. 13 (1982), 295–308.
  • H. W. Gould and L. C. Hsu, Some new inverse series relations, Duke Math. J. 40 (1973), 885–891.
  • D. H. Greene and D. E. Knuth, Mathematics for the Analysis of Algorithms. Basel: Birkhauser Verlag 1981.
  • C. Krattenthaler, A new matrix inverse. Preprint 1989.
  • A. Lakin, A hypergeometric identity related to Dougall’s theorem. J. London Math. Soc. 27 (1952), 229–234.
  • J. Riordan, Combinatorial Identities. New York: John-Wiley 1968.
  • L. J. Slater, Generalized Hypergeometric Functions. Cambridge: Cambridge Univ. Press 1966.
  • F. J. W. Whipple, On well-poised series, generalized hypergeometric series having parameters in pairs, each pair with the same sum. Proc....
  • H. S. Wilf and D. Zeilberger, Rational functions certify combinatorial identities. J. Amer. Math. Soc. 3:1, (1990), 147–158.