The Ramanujan Journal

, Volume 1, Issue 1, pp 7–23 | Cite as

The Well-Poised Thread: An Organized Chronicle of Some Amazing Summations and their Implications

  • George E. Andrews


This paper provides an organized history of well-poised hypergeometric series. The object is to reveal the process by which a rather narrow mathematical study blossomed into a topic of widespread importance. In addition, short biographies of the early contributors are included.

hypergeometric series well-poised series 


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  1. 1.
    G.E. Andrews, “On the general Rogers-Ramanujan theorem,” Memoirs of the Amer. Math. Soc. (152) (1974).Google Scholar
  2. 2.
    G.E. Andrews, “Applications of basic hypergeometric functions,” S.I.A.M. Review 16(1974), 441-484.Google Scholar
  3. 3.
    G.E. Andrews, “Problems and Prospects for Basic Hypergeometric Functions,” from Theory and Applications of Special Functions (R. Askey, ed.), Academic Press, New York, 1976, pp. 191-224.Google Scholar
  4. 4.
    G.E. Andrews, “The Theory of Partitions,” Encyclopedia of Math. and Its Applications, Addison-Wesley, Reading, Mass. vol. 2 (Reissued: Cambridge University Press, Cambridge, 1985).Google Scholar
  5. 5.
    G.E. Andrews, R.J. Baxter, and P.J. Forrester, “Eight-vertex SOS model and generalized Rogers-Ramanujantype identities,” J. Stat. Phys. 35(1984), 193-266.Google Scholar
  6. 6.
    G.E. Andrews, “Reciprocal polynomials and quadratic transformations,” Utilitas Math. 28(1985), 255-264.Google Scholar
  7. 7.
    G.E. Andrews, C. Bessenrodt, and J. Olsson, “Partition identities and labels for some modular characters,” Trans. Amer. Math. Soc. 344(1994), 597-615.Google Scholar
  8. 8.
    R. Askey, “Ramanujan's extension of the gamma and beta function,” Amer. Math. (Monthly) 87(1980), 346-359.Google Scholar
  9. 9.
    R. Askey, “Orthogonal Polynomials and Special Functions,” Regional Conf. Series in Applied Math. 21, S.I.A.M., Philadelphia.Google Scholar
  10. 10.
    W.N. Bailey, Generalized Hypergeometric Series, Cambridge University Press, 1935 (Reissued: Hafner, New York, 1964).Google Scholar
  11. 11.
    W.N. Bailey, “Francis John Welsh Whipple,” J. London Math. Soc. 18(1943), 249-256.Google Scholar
  12. 12.
    W.N. Bailey, “Some identities in combinatory analysis,” Proc. London Math. Soc. 49(2) (1947), 421-435.Google Scholar
  13. 13.
    W.N. Bailey, “Identities of the Rogers-Ramanujan type,” Proc. London Math. Soc. 50(2) (1949), 1-10.Google Scholar
  14. 14.
    B. Berndt, Ramanujan's Notebooks, Part III, Springer-Verlag, New York, 1991.Google Scholar
  15. 15.
    J.A. Daum, “The basic analog of Kummer's theorem,” Bull. Amer. Math. Soc. 48(1942), 711-713.Google Scholar
  16. 16.
    A.C. Dixon, “On the sum of the cubes of the coefficients in a certain expansion by the binomial theorem,” Mess. of Math. 20(1891), 79-80.Google Scholar
  17. 17.
    A.C. Dixon, “Leonard James Rogers,” Obit. Notices of Fellows of the Royal Society 1(1932-35), 299-301.Google Scholar
  18. 18.
    J. Dougall, “On Vandermonde's theorem and some more general expansions,” Proc. Edinburgh Math. Soc. 25(1907), 114-132.Google Scholar
  19. 19.
    G. Gasper and M. Rahman, “Basic hypergeometric series,” Encyclopedia of Math. and Its Applications, Cambridge University Press, Cambridge, 1990, vol. 35.Google Scholar
  20. 20.
    G.H. Hardy, Ramanujan, Cambridge University Press, Cambridge, 1940 (Reissued: Chelsea, New York, 1978).Google Scholar
  21. 21.
    F.H. Jackson, “Certain q-identities,” Quart. J. Math. Oxford Ser. 12(1941), 167-172.Google Scholar
  22. 22.
    E.E. Kummer, “Über die hypergeometrische Reihe...,” J. für die reine und angew. Math. 15(1836), 39-83, 127-172.Google Scholar
  23. 23.
    E.E. Kummer, Collected Papers (A. Weil, ed.), Springer-Verlag, Berlin, 1975, vols. I and II.Google Scholar
  24. 24.
    P.A. MacMahon, Collected Papers (G. Andrews, ed.), MIT Press, Cambridge, 1978, vol. 1.Google Scholar
  25. 25.
    S. Milne, “A q-analog of hypergeometric series well-poised in SU(n) and invariant G-functions,” Adv. in Math. 58(1985), 1-60.Google Scholar
  26. 26.
    F. Morley, “On the series...,” Proc. London Math. Soc. 34(1) (1902), 397-402.Google Scholar
  27. 27.
    H.W. Richmond, “The sum of the cubes of the coefficients in (1 x)2n,” Mess. of Math. 21(1892), 77-78.Google Scholar
  28. 28.
    J. Riordan, Combinatorial Identities, John Wiley, New York, 1968.Google Scholar
  29. 29.
    L.J. Rogers, “On a three-fold symmetry in the elements of Heine's series,” Proc. London Math. Soc. 24(1893), 171-179.Google Scholar
  30. 30.
    L.J. Rogers, “Second memoir on the expansion of certain infinite products,” Proc. London Math. Soc. 25(1) (1894), 318-343.Google Scholar
  31. 31.
    L.J. Rogers, “Third memoir on the expansion of certain infinite products,” Proc. London Math. Soc. 26(1), (1894), 15-32.Google Scholar
  32. 32.
    L.J. Slater, “W.N. Bailey,” J. London Math. Soc. 37(1962), 504-512.Google Scholar
  33. 33.
    G.N. Watson, “A new proof of the Rogers-Ramanujan identities,” J. London Math. Soc. 4(1929), 4-9.Google Scholar
  34. 34.
    F.J.W. Whipple, “On well-poised series, generalized hypergeometric series having parameters in pairs, each pair with the same sum,” Proc. London Math. Soc. 23(2) (1925), 104-114.Google Scholar
  35. 35.
    F.J.W. Whipple, “Some transformations of generalized hypergeometrics series,” Proc. London Math. Soc. 26(2), (1927), 257-272.Google Scholar
  36. 36.
    E.T. Whittaker, “Alfred Carden Dixon,” J. London Math. Soc. 12(1937), 145-155.Google Scholar
  37. 37.
    J.M. Whittaker, “George Neville Watson,” Biographical Memoirs of the Royal Soc. 12(1966), 521-530.Google Scholar

Copyright information

© Kluwer Academic Publishers 1997

Authors and Affiliations

  • George E. Andrews
    • 1
  1. 1.Department of Mathematics, Eberly College of SciencesThe Pennsylvania State UniversityUniversity ParkPennsylvania

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