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A simple proof of Bailey's very-well-poised ${}_{6}\psi_{6}$ summation

Author: Michael Schlosser
Journal: Proc. Amer. Math. Soc. 130 (2002), 1113-1123
MSC (2000): Primary 33D15
Published electronically: October 1, 2001
MathSciNet review: 1873786
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Abstract: We give elementary derivations of some classical summation formulae for bilateral (basic) hypergeometric series. In particular, we apply Gauß' $_2F_1$ summation and elementary series manipulations to give a simple proof of Dougall's $_2H_2$ summation. Similarly, we apply Rogers' nonterminating $_6\phi_5$ summation and elementary series manipulations to give a simple proof of Bailey's very-well-poised $_6\psi_6$ summation. Our method of proof extends M. Jackson's first elementary proof of Ramanujan's $_1\psi_1$ summation.

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  • 1. G. E. Andrews, Applications of basic hypergeometric functions, SIAM Rev. 16 (1974), 441-484. MR 50:5044
  • 2. -, private communication, June 2000.
  • 3. G. E. Andrews, R. Askey and R. Roy, Special functions, Encyclopedia of Mathematics And Its Applications 71, Cambridge University Press, Cambridge (1999). MR 2000g:33001
  • 4. R. Askey, The very well poised $_6\psi_6$. II, Proc. Amer. Math. Soc. 90 (1984), 575-579. MR 85h:33001
  • 5. R. Askey and M. E. H. Ismail, The very well poised $_6\psi_6$, Proc. Amer. Math. Soc. 77 (1979), 218-222. MR 80m:33002
  • 6. W. N. Bailey, Series of hypergeometric type which are infinite in both directions, Quart. J. Math. (Oxford) 7 (1936), 105-115.
  • 7. -, On the basic bilateral basic hypergeometric series ${}_2\psi_2$, Quart. J. Math. (Oxford) (2) 1 (1950), 194-198. MR 12:178e
  • 8. A.-L. Cauchy, Mémoire sur les fonctions dont plusieurs valeurs $\ldots$, C. R. Acad. Sci. Paris 17 (1843), 523; reprinted in Oeuvres de Cauchy, Ser. 1 8, Gauthier-Villars, Paris (1893), 42-50.
  • 9. J. F. van Diejen, On certain multiple Bailey, Rogers and Dougall type summation formulas, Publ. Res. Inst. Math. Sci., Ser. A 33 (1997), 483-508. MR 98j:33011
  • 10. J. Dougall, On Vandermonde's theorem and some more general expansions, Proc. Edinburgh Math. Soc. 25 (1907), 114-132.
  • 11. G. Gasper and M. Rahman, Basic hypergeometric series, Encyclopedia of Mathematics And Its Applications 35, Cambridge University Press, Cambridge (1990). MR 91d:33034
  • 12. C. F. Gauß, Disquisitiones generales circa seriem infinitam $\ldots$, Comm. soc. reg. sci. Gött. rec. 2 (1813), reprinted in his Werke (Göttingen), vol. 3 (1860), 123-163.
  • 13. R. A. Gustafson, The Macdonald identities for affine root systems of classical type and hypergeometric series very well-poised on semi-simple Lie algebras, in Ramanujan International Symposium on Analysis (Dec. 26th to 28th, 1987, Pune, India), N. K. Thakare (ed.) (1989), 187-224. MR 92k:33015
  • 14. W. Hahn, Beiträge zur Theorie der Heineschen Reihen, Die 24 Integrale der hypergeometrischen $q$-Differenzengleichung, Das $q$-Analogon der Laplace-Transformation, Math. Nachr. 2 (1949), 340-379. MR 11:720b
  • 15. G. H. Hardy, Ramanujan, Cambridge University Press, Cambridge (1940), reprinted by Chelsea, New York, 1978. MR 3:71 (original review)
  • 16. E. Heine, Untersuchungen über die Reihe $\ldots$, J. reine angew. Math. 34 (1847), 285-328.
  • 17. M. E. H. Ismail, A simple proof of Ramanujan's $_1\psi_1$sum, Proc. Amer. Math. Soc. 63 (1977), 185-186. MR 58:22695
  • 18. F. H. Jackson, Summation of $q$-hypergeometric series, Messenger of Math. 57 (1921), 101-112.
  • 19. M. Jackson, On Lerch's transcendant and the basic bilateral hypergeometric series $_2\psi_2$, J. London Math. Soc. 25 (1950), 189-196. MR 12:178f
  • 20. T. H. Koornwinder, On Zeilberger's algorithm and its $q$-analogue, J. Comp. and Appl. Math. 48 (1993), 91-111. MR 95b:33011
  • 21. S. C. Milne, Balanced $_3\phi_2$ summation theorems for $U(n)$ basic hypergeometric series, Adv. Math. 131 (1997), 93-187. MR 99d:33025
  • 22. R. J. Rogers, Third memoir on the expansion of certain infinite products, Proc. London Math. Soc. 26 (1894), 15-32.
  • 23. M. Schlosser, Summation theorems for multidimensional basic hypergeometric series by determinant evaluations, Discrete Math. 210 (2000), 151-169. CMP 2000:08
  • 24. -, Elementary derivations of identities for bilateral basic hypergeometric series, preprint.
  • 25. L. J. Slater, General transformations of bilateral series, Quart. J. Math. (Oxford) (2) 3 (1952), 73-80. MR 14:271b
  • 26. -, Generalized hypergeometric functions, Cambridge Univ. Press, London/New York, 1966. MR 34:1570
  • 27. L. J. Slater and A. Lakin, Two proofs of the $_6\psi_6$summation theorem, Proc. Edinburgh Math. Soc. (2) 9 (1953-57), 116-121. MR 18:888b

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Additional Information

Michael Schlosser
Affiliation: Department of Mathematics, The Ohio State University, 231 West 18th Avenue, Columbus, Ohio 43210
Address at time of publication: Institut für Mathematik der Universität Wien, Strudlhofgasse 4, A-1090 Wien, Austria

Keywords: Bilateral basic hypergeometric series, $q$-series, Ramanujan's $_1\psi_1$ summation, Dougall's $_2H_2$ summation, Bailey's $_6\psi_6$ summation
Received by editor(s): July 7, 2000
Received by editor(s) in revised form: September 25, 2000, and October 18, 2000
Published electronically: October 1, 2001
Communicated by: John R. Stembridge
Article copyright: © Copyright 2001 American Mathematical Society

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