Advertisement

Hypergeometric Series

  • George E. Andrews
  • Bruce C. Berndt
Chapter

Abstract

Two fascinating formulas for bilateral hypergeometric series are proved. The second portion of the chapter is devoted to a beautiful continued fraction related to hypergeometric polynomials.

Keywords

Continue Fraction Moment Problem Wilson Polynomial Lost Notebook Hahn Polynomial 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 10.
    G. Andrews and R. Askey, Classical orthogonal polynomials, in Polynômes Orthogonaux et Applications, Lecture Notes in Mathematics No. 1171, Springer-Verlag, New York, 1985, pp. 36–82.Google Scholar
  2. 11.
    G.E. Andrews, R. Askey, and R. Roy, Special Functions, Cambridge University Press, Cambridge, 2000.MATHGoogle Scholar
  3. 16.
    R. Askey and J.A. Wilson, A set of hypergeometric orthogonal polynomials, SIAM J. Math. Anal. 13 (1982), 651–655.CrossRefMathSciNetMATHGoogle Scholar
  4. 17.
    R. Askey and J.A. Wilson, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc. 319, 1985.Google Scholar
  5. 22.
    E.W. Barnes, A new development of the theory of the hypergeometric functions, Proc. London Math. Soc. (2) 6 (1908), 141–177.Google Scholar
  6. 38.
    B.C. Berndt, Ramanujan’s Notebooks, Part II, Springer-Verlag, New York, 1989.CrossRefMATHGoogle Scholar
  7. 50.
    B.C. Berndt and W. Chu, Two entries on bilateral hypergeometric series in Ramanujan’s lost notebook, Proc. Amer. Math. Soc. 135 (2007), 129–134.CrossRefMathSciNetMATHGoogle Scholar
  8. 113.
    J. Dougall, On Vandermonde’s theorem and some more general expansions, Proc. Edinburgh Math. Soc. 25 (1907), 114–132.CrossRefMATHGoogle Scholar
  9. 124.
    M.L. Glasser, Evaluation of a class of definite integrals, Univ. Beograd Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 506 (1975), 49–50.Google Scholar
  10. 164.
    M.E. Horn, Bilateral binomial theorem, SIAM Problem 03-001 (2003), solutions by J.M. Borwein and M.E. Horn, and by G.C. Greubel, http://www.siam.org/journals/categories/03-001.php.
  11. 165.
    M.E.H. Ismail, J. Letessier, G. Valent, and J. Wimp, Two families of associated Wilson polynomials, Canad. J. Math. 42 (1990), 659–695.CrossRefMathSciNetMATHGoogle Scholar
  12. 175.
    S.-Y. Kang, S.-G. Lim, and J. Sohn, The continuous symmetric Hahn polynomials found in Ramanujan’s lost notebook, J. Math. Anal. Appl. 307 (2005), 153–166.CrossRefMathSciNetMATHGoogle Scholar
  13. 187.
    T.H. Koornwinder, Compact quantum groups and q-special functions, in Representations of Lie groups and quantum groups, V. Baldoni and M.A. Picardello, eds., Longman, Harlow, Essex, UK, 1994, pp. 46–128Google Scholar
  14. 218.
    L. Lorentzen and H. Waadeland, Continued Fractions, Vol. 1: Convergence Theory, World Scientific Press, Singapore, 2008.Google Scholar
  15. 255.
    S. Ramanujan, Some definite integrals, Mess. Math. 44 (1915), 10–18.Google Scholar
  16. 267.
    S. Ramanujan, Collected Papers, Cambridge University Press, Cambridge, 1927; reprinted by Chelsea, New York, 1962; reprinted by the American Mathematical Society, Providence, RI, 2000.Google Scholar
  17. 269.
    S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa, New Delhi, 1988.MATHGoogle Scholar
  18. 273.
    R. Roy, On a paper of Ramanujan on definite integrals, Math. Student 46 (1978), 130–132.MathSciNetGoogle Scholar
  19. 286.
    J.A. Shohat and J.D. Tamarkin, The Problem of Moments, American Mathematical Society, Providence, RI, 1943.CrossRefMATHGoogle Scholar
  20. 290.
    L.J. Slater, Generalized Hypergeometric Functions, Cambridge University Press, Cambridge, 1966.MATHGoogle Scholar
  21. 297.
    T.J. Stieltjes, Sur quelques intégrales définies et leur développement en fractions continues, Quart. J. Math. 24 (1890), 370–382.MATHGoogle Scholar
  22. 298.
    T.J. Stieltjes, Oeuvres, Vol. 2, Noordhoff, Groningen, 1918, pp. 378–394.MATHGoogle Scholar
  23. 305.
    E.C. Titchmarsh, Theory of Fourier Integrals, Clarendon Press, Oxford, 1937; 3rd ed., Chelsea, New York, 1986.Google Scholar
  24. 312.
    H.S. Wall, Analytic Theory of Continued Fractions, D. van Nostrand, New York, 1948; reprinted by Chelsea, New York, 1967; reprinted by the American Mathematical Society, Providence, RI, 2000.Google Scholar
  25. 318.
    J.A. Wilson, Hypergeometric Series, Recurrence Relations and Some New Orthogonal Functions, Ph.D. Thesis, University of Wisconsin, Madison, 1978.Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • George E. Andrews
    • 1
  • Bruce C. Berndt
    • 2
  1. 1.Department of MathematicsThe Pennsylvania State UniversityUniversity ParkUSA
  2. 2.Department of MathematicsUniversity of Illinois at Urbana-ChampaignUrbanaUSA

Personalised recommendations