Summation Theorems for Basic Hypergeometric Series of Schur Function Argument

  • S. C. Milne
Conference paper
Part of the Springer Series in Computational Mathematics book series (SSCM, volume 19)


In this paper we prove a Ramanujan 1 ψ 1 summation theorem for a Laurent series extension of I.G. Macdonald’s (Schur function) multiple basic hypergeometric series of matrix argument. This result contains as special, limiting cases our Schur function extension of the q-binomial theorem and the Jacobi triple product identity. Just as in the classical case, we write our new q-binomial theorem and 1 ψ 1 summation as elegant special cases of K. Kadell’s and R. Askey’s q-analogs of Selberg’s multiple beta-integral. We also apply our q-binomial theorem and K. Kadell’s Schur function q-analog of Selberg’s beta-integral to derive a Heine transformation and q-Gauss summation theorem for Schur functions.


Symmetric Function Hypergeometric Series Vandermonde Determinant Basic Hypergeometric Series Matrix Argument 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    G.E. Andrews, “The Theory of Partitions”, Vol. 2, “Encyclopedia of Mathematics and Its Applications”, (G.-C. Rota, Ed.), Addison-Wesley, Reading, Mass., 1976.Google Scholar
  2. [2]
    G.E. Andrews, “q-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra”, CBMS Regional Conference Lecture Series 66 (1986), Amer. Math. Soc, Providence, R.I.Google Scholar
  3. [3]
    G.E. Andrews, On Ramanujan’s summation of 1 ψ 1(a, b, z), Proc. Amer. Math. Soc. 22 (1969), 552–553.MathSciNetzbMATHGoogle Scholar
  4. [4]
    G.E. Andrews, On a transformation of bilateral series with applications, Proc. Amer. Math. Soc, 25 (1970), 554–558.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    G.E. Andrews, Connection coefficient problems and partitions, AMS Proc. Sympos. Pure Math. 34 (1979), 1–24.Google Scholar
  6. [6]
    G.E. Andrews and R. Askey, “The Classical and Discrete Orthogonal Polynomials and Their q-Analogues”, in preparation.Google Scholar
  7. [7]
    G.E. Andrews and R. Askey, Enumeration of partitions: the role of Eulerian series and q-orthogonal polynomials, “Higher Combinatorics” (M. Aigner, ed.) Reidel, Boston, (1977), p. 3–26.Google Scholar
  8. [8]
    G.E. Andrews and R. Askey, A simple proof of Ramanujan’s summation of the 1 ψ 1, Aequationes Math. 18 (1978), 333–337.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    G.E. Andrews and R. Askey, Another q-extension of the beta function, Proc. Amer. Math. Soc. 81 (1981), 97–100.MathSciNetzbMATHGoogle Scholar
  10. [10]
    G.E. Andrews and R. Askey, Classical orthogonal polynomials, “Polynomes Orthogonaux et Applications”, Lecture Notes in Math. 1171 (1985), Springer, Berlin and New York, p. 36–62.CrossRefGoogle Scholar
  11. [11]
    G.E. Andrews, R. Askey, B.C. Berndt, K.G. Ramanathan, and R.A. Rankin, eds., “Ramanujan Revisited”, Academic Press, New York (1988).zbMATHGoogle Scholar
  12. [12]
    K. Aomoto, Jacobi polynomials associated with Selberg’s integral, SIAM J. Math. Anal. 18 (1987), 545–549.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    R. Askey, The q-gamma and q-beta functions, Appl. Anal. 8 (1978), 125–141.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [14]
    R. Askey, Ramanujan’s extensions of the gamma and beta functions, Amer. Math. Monthly 87 (1980), 346–359.MathSciNetzbMATHCrossRefGoogle Scholar
  15. [15]
    R. Askey, Some basic hypergeometric extensions of integrals of Selberg and Andrews, SIAM J. Math. Anal. 11 (1980), 938–951.MathSciNetzbMATHCrossRefGoogle Scholar
  16. [16]
    R. Askey, Two integrals of Ramanujan, Proc. Amer. Math. Soc. 85 (1982), 192–194.MathSciNetzbMATHCrossRefGoogle Scholar
  17. [17]
    R. Askey, A q-beta integral associated with BC 1, SIAM J. Math. Anal. 13 (1982), 1008–1010.MathSciNetzbMATHCrossRefGoogle Scholar
  18. [18]
    R. Askey, An elementary evaluation of a beta type integral, Indian J. Pure Appl. Math. 14 (1983), 892–895.MathSciNetzbMATHGoogle Scholar
  19. [19]
    R. Askey, Orthogonal polynomials old and new, and some combinatorial connections, “Enumeration and Design” (D.M. Jackson and S.A. Vanstone, eds.,), Academic Press, New York (1984), p. 67–84.Google Scholar
  20. [20]
    R. Askey, Ramanujan’s 1 ψ 1 and formal Laurent series, Indian J. Math. 29 (1987), 101–105.MathSciNetzbMATHGoogle Scholar
  21. [21]
    R. Askey, Beta integrals in Ramanujan’s papers, his unpublished work and further examples, “Ramanujan Revisited” (G.E. Andrews et al., eds.), Academic Press, New York (1988), p. 561–590.Google Scholar
  22. [22]
    R. Askey and R. Roy, More q-beta integrals, Rocky Mountain J. Math. 16 (1986), 365–372.MathSciNetzbMATHCrossRefGoogle Scholar
  23. [23]
    R. Askey and J. Wilson, A set of orthogonal polynomials that generalize the Racah coefficients or 6 — j symbols, SIAM J. Math. Anal. 10 1979), 1008–1016.MathSciNetzbMATHCrossRefGoogle Scholar
  24. [24]
    W.N. Bailey, “Generalized Hypergeometric Series”, Cambridge Mathematical Tract No. 32, Cambridge University Press, Cambridge, 1935.zbMATHGoogle Scholar
  25. [25]
    N.J. Fine, “Basic Hypergeometric Series and Applications”, Mathematical Surveys and Monographs, Vol. 27 (1988), Amer. Math. Soc, Providence, R.I.zbMATHGoogle Scholar
  26. [26]
    G. Gasper and M. Rahman, “Basic Hypergeometric Series”, Vol. 35, “Encyclopedia of Mathematics and Its Applications”, (G.-C. Rota, Ed.), Cambridge University Press, Cambridge, 1990.Google Scholar
  27. [27]
    S.G. Gindikin, Analysis on homogeneous spaces, Russian Math. Surveys 19 (1964), 1–90.MathSciNetCrossRefGoogle Scholar
  28. [28]
    K.I. Gross and D. St. P. Richards, Special functions of matrix argument I: Alebraic induction, zonal polynomials, and hypergeometric functions, Trans. Amer. Math. Soc. 301 (1987), 781–811.MathSciNetzbMATHGoogle Scholar
  29. [29]
    W. Hahn, Über orthogonal polynome, die Differenzengleichungen genügen, Math. Nachr 2 (1949), 4–34.MathSciNetzbMATHCrossRefGoogle Scholar
  30. [30]
    W. Hahn, Beitrage zur Theorie der Heineschen Reihen, Math. Nachr. 2 (1949), 340–379.MathSciNetzbMATHCrossRefGoogle Scholar
  31. [31]
    P. Hall, The algebra of partitions, in Proceedings, 4th Canadian Math. Congress, Banff (1959), p. 147–159.Google Scholar
  32. [32]
    E. Heine, Untersuchungen über die Reihe..., J. Reine Angew. Math. 34 (1847), 285–328.zbMATHCrossRefGoogle Scholar
  33. [33]
    C.S. Herz, Bessel functions of matrix argument, Ann. of Math. (2) 61 (1955), 474–523.MathSciNetzbMATHCrossRefGoogle Scholar
  34. [34]
    M.E.H. Ismail, A simple proof of Ramanujan’s 1 ψ 1 sum, Proc. Amer. Math. Soc. 63 (1977), 185–186.MathSciNetzbMATHGoogle Scholar
  35. [35]
    F.H. Jackson, Transformations of q-series, Messenger of Math. 39 (1910), 145–153.Google Scholar
  36. [36]
    M. Jackson, On Lerch’s transcendent and the basic bilateral hypergeometric series 2 ψ 2, J. London Math. Soc. 25 (1950), 189–196.MathSciNetzbMATHCrossRefGoogle Scholar
  37. [37]
    C.G.J. Jacobi, “Fundamenta nova theoriae functionum ellipticarum”, (1829), Rigiomnoti, fratrum Bornträger (reprinted in “Gesammelte Werke”, Vol. 1, pp. 49–239, Reimer, Berlin, 1881).Google Scholar
  38. [38]
    C.G. Jacobi, De functionibus alternantibus..., Crelle’s Journal 22 (1841), 360–371 [Werke 3, 439–452].zbMATHCrossRefGoogle Scholar
  39. [39]
    A.T. James, Distribution of matrix variates and latent roots derived from normal samples, Ann. Math. Statist. 35 (1964), 475–501.MathSciNetzbMATHCrossRefGoogle Scholar
  40. [40]
    K. Kadell, A proof of some q-analogs of Selberg’s integral for k = 1, SIAM J. Math. Analysis 19 (1988), pp. 944–968.MathSciNetzbMATHCrossRefGoogle Scholar
  41. [41]
    K. Kadell, The Selberg-Jack polynomials, to appear.Google Scholar
  42. [42]
    D.E. Littlewood, “The Theory of Group Characters”, 2nd ed. Oxford at the Clarendon Press, 1940.Google Scholar
  43. [43]
    D.E. Littlewood and A.R. Richardson, Group characters and algebra, Philos. Trans. Roy. Soc. London Ser. A 233 (1934), 99–141.CrossRefGoogle Scholar
  44. [44]
    J.D. Louck and L.C. Biedenharn, A generalization of the Gauss hypergeometric function, J. Math. Anal. Appl. 59 (1977), 423–431.MathSciNetzbMATHCrossRefGoogle Scholar
  45. [45]
    I.G. Macdonald, “Symmetric Functions and Hall Polynomials”, Oxford Univ. Press, London/New York, 1979.zbMATHGoogle Scholar
  46. [46]
    I.G. Macdonald, A new class of symmetric functions, Publ. I.R.M.A. Strasbourg, 1988, 372/s-20, Actes 20e Séminaire Lotharingien, p. 131–171.Google Scholar
  47. [47]
    I.G. Macdonald, Lecture notes from his talk, Univer. of Michigan, June 1989.Google Scholar
  48. [48]
    S.C. Milne, A U(n) generalization of Ramanujan’s 1 ψ 1 summation, J. Math. Anal. Appl. 118 (1986), 263–277.MathSciNetzbMATHCrossRefGoogle Scholar
  49. [49]
    S.C. Milne, Multiple q-series and U(n) generalizations of Ramanujan’s 1Ψ1 sum, “Ramanujan Revisited”, (G.E. Andrews et al. eds.), Academic Press, New York (1988), p. 473–524.Google Scholar
  50. [50]
    S.C. Milne, The multidimensional 1Ψ1 sum and Macdonald identities for Al(1), Proc. Sympos. Pure Math. 49 (part 2)(1989), 323–359.MathSciNetGoogle Scholar
  51. [51]
    S.C. Milne, A triple product identity for Schur functions, J. Math. Anal. Appl., in press.Google Scholar
  52. [52]
    M. Rahman, Some extensions of the beta integral and the hypergeometric function, to appear.Google Scholar
  53. [53]
    E.D. Rainville, “Special Functions”, Macmillan Co., New York, 1960.zbMATHGoogle Scholar
  54. [54]
    J. Remmel and R. Whitney, Multiplying Schur functions, J. Algorithms 5 (1984), 471–487.MathSciNetzbMATHCrossRefGoogle Scholar
  55. [55]
    I. Schur, “Über ein Klasse von Matrizen die sich einer gegebenen Matrix zuordnen lassen”, Dissertation, Berlin, 1901. [Ges Abhandlungen I, 1–72].Google Scholar
  56. [56]
    M.-P. Schützenberger, La correspondance de Robinson, in “Combinatoire et represéntation du groupe symétrique”, Strasbourg, 1976. Lecture Notes in Mathematics No. 579, Springer-Verlag, New York/Berlin, 1977.Google Scholar
  57. [57]
    A. Seiberg, Bemerkninger om et multpelt integral, Norske Mat. Tidsskr. 26 (1944), 71–78.Google Scholar
  58. [58]
    D.P. Shukla, Certain transformations of generalized hypergeometric series, Indian J. Pure Appl. Math. 12 (8) (1981), 994–1000.MathSciNetzbMATHGoogle Scholar
  59. [59]
    L.J. Slater, “Generalized Hypergeometric Functions”, Cambridge University Press, London and New York, 1966.zbMATHGoogle Scholar
  60. [60]
    R.P. Stanley, Theory and applications of plane partitions, Studies in Applied Mathematics 50 (1971), 167–188, 259–279.MathSciNetzbMATHGoogle Scholar
  61. [61]
    J. Thomae, Beiträge zur Theorie der durch die Heinesche Reihe..., J. Reine Angew. Math. 70 (1869), 258–281.CrossRefGoogle Scholar
  62. [62]
    J. Thomae, Les séries Heinéennes supérieures, ou les séries de la forme..., Annali di Matematica Pura de Applicata 4 (1870), 105–138.CrossRefGoogle Scholar
  63. [63]
    J.A. Wilson, “Hypergeometric Series, Recurrence Relations and Some New Orthogonal Polynomials”, Thesis (1978), Univ. of Wisconsin, Madison.Google Scholar

Copyright information

© Springer-Verlag New York, Inc. 1992

Authors and Affiliations

  • S. C. Milne
    • 1
  1. 1.Department of MathematicsOhio State UniversityColumbusUSA

Personalised recommendations