Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 3 (2007), 056, 30 pages      math.CO/0611639      https://doi.org/10.3842/SIGMA.2007.056
Contribution to the Vadim Kuznetsov Memorial Issue

Macdonald Polynomials and Multivariable Basic Hypergeometric Series

Michael J. Schlosser
Fakultät für Mathematik, Universität Wien, Nordbergstraàe 15, A-1090 Vienna, Austria

Received November 21, 2006; Published online March 30, 2007

Abstract
We study Macdonald polynomials from a basic hypergeometric series point of view. In particular, we show that the Pieri formula for Macdonald polynomials and its recently discovered inverse, a recursion formula for Macdonald polynomials, both represent multivariable extensions of the terminating very-well-poised 6φ5 summation formula. We derive several new related identities including multivariate extensions of Jackson's very-well-poised 8φ7 summation. Motivated by our basic hypergeometric analysis, we propose an extension of Macdonald polynomials to Macdonald symmetric functions indexed by partitions with complex parts. These appear to possess nice properties.

Key words: Macdonald polynomials; Pieri formula; recursion formula; matrix inversion; basic hypergeometric series; 6φ5 summation; Jackson's 8φ7 summation; An-1 series.

pdf (455 kb)   ps (292 kb)   tex (33 kb)

References

  1. Andrews G.E., q-series: their development and application in analysis, number theory, combinatorics, physics and computer algebra, CBMS Regional Conference Lectures Series, Vol. 66, Amer. Math. Soc., Providence, RI, 1986.
  2. Andrews G.E., Askey R., Roy R., Special functions, Encyclopedia of Mathematics and Its Applications, Vol. 71, Cambridge University Press, Cambridge, 1999.
  3. Baker T.H., Forrester P.J., Transformation formulas for multivariable basic hypergeometric series, Meth. Appl. Anal. 6 (1999), 147-164, math.QA/9803146.
  4. Bhatnagar G., A multivariable view of q-series, in Special Functions and Differential Equations, Editors K. Srinivasa Rao, R. Jagannathan, G. Vanden Berghe and J. Van der Jeugt, Proceedings of a Workshop, WSSF'97 (January 13-24, 1997, Madras, India), Allied Publ., New Delhi, 1998, 25-30.
  5. Bhatnagar G., Milne S.C., Generalized bibasic hypergeometric series and their U(n) extensions, Adv. Math. 131 (1997), 188-252.
  6. Bressoud D.M., A matrix inverse, Proc. Amer. Math. Soc. 88 (1983), 446-448.
  7. Bromwich T.J.l'A., An introduction to the theory of infinite series, 2nd ed., Macmillan, London, 1949.
  8. Cauchy A.-L., Mémoire sur les fonctions dont plusieurs valeurs sont liées entre elles par une équation linéaire, et sur diverses transformations de produits composés d'un nombre indéfini de facteurs, C. R. Acad. Sci. Paris XVII (1843), 523; Oeuvres de Cauchy, 1re série, T. VIII, Gauthier-Villars, Paris, 1893, 42-50.
  9. Danilov V., Koshevoy G., Continuous combinatorics, Preprint, 2005, 12 pages.
  10. Denis R.Y., Gustafson R.A., An SU(n) q-beta integral transformation and multiple hypergeometric series identities, SIAM J. Math. Anal. 23 (1992), 552-561.
  11. Frenkel I.B., Turaev V.G., Elliptic solutions of the Yang-Baxter equation and modular hypergeometric functions, in The Arnold-Gelfand Mathematical Seminars, Editors V.I. Arnold, I.M. Gelfand, V.S. Retakh and M. Smirnov, Birkhäuser, Boston, 1997, 171-204.
  12. Gasper G., Rahman M., Basic hypergeometric series, 2nd ed., Encyclopedia of Mathematics and Its Applications, Vol. 96, Cambridge University Press, Cambridge, 2004.
  13. Gustafson R.A., Multilateral summation theorems for ordinary and basic hypergeometric series in U(n), SIAM J. Math. Anal. 18 (1987), 1576-1596.
  14. Gustafson R.A., 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 (December 26-28, 1987, Pune, India), Editor N.K. Thakare, 1989, 187-224.
  15. Heine E., Untersuchungen über die Reihe ¼, J. Reine Angew. Math. 34 (1847), 285-328.
  16. Holman W.J. III, Summation theorems for hypergeometric series in U(n), SIAM J. Math. Anal. 11 (1980), 523-532.
  17. Holman W.J. III, Biedenharn L.C., Louck J.D., On hypergeometric series well-poised in SU(n), SIAM J. Math. Anal. 7 (1976), 529-541.
  18. Ismail M.E.H., Classical and quantum orthogonal polynomials in one variable, Encyclopedia of Mathematics and its Applications, Vol. 98, Cambridge University Press, Cambridge, 2005.
  19. Jackson F.H., Summation of q-hypergeometric series, Messenger of Math. 57 (1921), 101-112.
  20. Jing N.H., Józefiak T., A formula for two-row Macdonald functions, Duke Math. J. 67 (1992), 377-385.
  21. Kadell K., The Schur functions for partitions with complex parts, Contemp. Math. 254 (2000), 247-270.
  22. Kajihara Y., Noumi M., Raising operators of row type for Macdonald polynomials, Compos. Math. 120 (2000), 119-136, math.QA/9803151.
  23. Kaneko J., q-Selberg integrals and Macdonald polynomials, Ann. Sci. École Norm. Sup. 29 (1996), 583-637.
  24. Kaneko J., A 1Y1 summation theorem for Macdonald polynomials, Ramanujan J. 2 (1998), 379-386.
  25. Koornwinder T.H., Self-duality for q-ultraspherical polynomials associated with root system An, unpublished handwritten manuscript, 1988, 17 pages.
    Available at http://remote.science.uva.nl/~thk/art/informal/dualmacdonald.pdf.
  26. Krattenthaler C., A new matrix inverse, Proc. Amer. Math. Soc. 124 (1996), 47-59.
  27. Krattenthaler C., Schlosser M.J., A new multidimensional matrix inverse with applications to multiple q-series, Discrete Math. 204 (1999), 249-279.
  28. Lassalle M., Explicitation des polynômes de Jack et de Macdonald en longueur trois, C. R. Acad. Sci. Paris Sér. I Math. 333 (2001), 505-508.
  29. Lassalle M., Une q-spécialisation pour les fonctions symétriques monomiales, Adv. Math. 162 (2001), 217-242, math.CO/0004019.
  30. Lassalle M., A short proof of generalized Jacobi-Trudi expansions for Macdonald polynomials, Contemp. Math. 417 (2006), 271-280, math.CO/0401032.
  31. Lassalle M., Schlosser M.J., Inversion of the Pieri formula for Macdonald polynomials, Adv. Math. 202 (2006), 289-325, math.CO/0402127.
  32. Macdonald I.G., A new class of symmetric functions, Sém. Lothar. Combin. 20 (1988), Art. B20a, 41 pages.
  33. Macdonald I.G., Symmetric functions and hall polynomials, 2nd ed., Clarendon Press, Oxford, 1995.
  34. Macdonald I.G., Symmetric functions and orthogonal polynomials, Dean Jacqueline B. Lewis Memorial Lectures presented at Rutgers University, New Brunswick, NJ, University Lecture Series, Vol. 12, Amer. Math. Soc., Providence, RI, 1998.
  35. Milne S.C., An elementary proof of the Macdonald identities for Al(1), Adv. Math. 57 (1985), 34-70.
  36. Milne S.C., Basic hypergeometric series very well-poised in U(n), J. Math. Anal. Appl. 122 (1987), 223-256.
  37. Milne S.C., Multiple q-series and U(n) generalizations of Ramanujan's 1y1 sum, in Ramanujan Revisited, Editors G.E. Andrews et al., Academic Press, New York, 1988, 473-524.
  38. Milne S.C., The multidimensional 1Y1 sum and Macdonald identities for Al(1), in Theta Functions Bowdoin (1987), Editors L. Ehrenpreis and R.C. Gunning, Proc. Sympos. Pure Math. 49 (1989), 323-359.
  39. Milne S.C., A q-analog of a Whipple's transformation for hypergeometric series in U(n), Adv. Math. 108 (1994), 1-76.
  40. Milne S.C., Balanced 3f2 summation theorems for U(n) basic hypergeometric series, Adv. Math. 131 (1997), 93-187.
  41. Milne S.C., Transformations of U(n+1) multiple basic hypergeometric series, in Physics and Combinatorics, Editors A.N. Kirillov, A. Tsuchiya and H. Umemura, Proceedings of the Nagoya 1999 International Workshop (August 23-27, 1999, Nagoya University, Japan), World Scientific, Singapore, 2001, 201-243.
  42. Milne S.C., Lilly G.M., Consequences of the Al and Cl Bailey transform and Bailey lemma, Discrete Math. 139 (1995), 319-346, math.CA/9204236.
  43. Milne S.C., Newcomb J.W., U(n) very-well-poised 10f9 transformations, J. Comput. Appl. Math. 68 (1996), 239-285.
  44. Okounkov A., BC-type interpolation Macdonald polynomials and binomial formula for Koornwinder polynomials, Transform. Groups 3 (1998), 181-207, q-alg/9611011.
  45. Rains E.M., BCn-symmetric polynomials, Transform. Groups 10 (2005), 63-132, math.QA/0112035.
  46. Rains E.M., BCn-symmetric Abelian functions, Duke Math. J. 135 (2006), 99-180, math.CO/0402113.
  47. Rogers R.J., Third memoir on the expansion of certain infinite products, Proc. London Math. Soc. 26 (1894), 15-32.
  48. Rosengren H., Elliptic hypergeometric series on root systems, Adv. Math. 181 (2004), 417-447, math.CA/0207046.
  49. Rosengren H., Reduction formulas for Karlsson-Minton-type hypergeometric functions, Constr. Approx. 20 (2004), 525-548, math.CA/0202232.
  50. Schlosser M.J., Multidimensional matrix inversions and Ar and Dr basic hypergeometric series, Ramanujan J. 1 (1997), 243-274.
  51. Schlosser M.J., A new multidimensional matrix inversion in Ar, Contemp. Math. 254 (2000), 413-432.
  52. Spiridonov V.P., Elliptic hypergeometric functions, Dr.Sc. Thesis, JINR, Dubna, Russia, 2004, 218 pages.
  53. Stanton D., An elementary approach to the Macdonald identities, in q-Series and Partitions, Editor D. Stanton, The IMA Volumes in Mathematics and Its Applications, Vol. 18, Springer-Verlag, 1989, 139-150.
  54. Warnaar S.O., q-Selberg integrals and Macdonald polynomials, Ramanujan J. 10 (2005), 237-268.
  55. Watson G.N., A treatise on the theory of Bessel functions, 2nd ed., Cambridge University Press, Cambridge, 1966.

Previous article   Next article   Contents of Volume 3 (2007)