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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)

Abstract

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.

Keywords

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

  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

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