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$ q$-Hermite polynomials, biorthogonal rational functions, and $ q$-beta integrals


Authors: M. E. H. Ismail and D. R. Masson
Journal: Trans. Amer. Math. Soc. 346 (1994), 63-116
MSC: Primary 33D15; Secondary 33D05, 33D45
DOI: https://doi.org/10.1090/S0002-9947-1994-1264148-6
MathSciNet review: 1264148
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Abstract: We characterize the solutions of the indeterminate moment problem associated with the continuous $ q$-Hermite polynomials when $ q > 1$ in terms of their Stieltjes transforms. The extremal measures are found explicitly. An analog of the Askey-Wilson integral is evaluated. It involves integrating a kernel, similar to the Askey-Wilson kernel, against any solution of the $ q$-Hermite moment problem, provided that certain integrability conditions hold. This led to direct evaluation of several $ q$-beta integrals and their discrete analogs as well as a generalization of Bailey's $ {}_6{\psi _6}$, sum containing infinitely many parameters. A system of biorthogonal rational functions is also introduced.


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  • [1] N. I. Akhiezer, The classical moment problem and some related questions in analysis, English transl., Oliver and Boyd, Edinburgh, 1965. MR 0184042 (32:1518)
  • [2] W. A. Al-Salam and L. Carlitz, Some orthogonal $ q$-polynomials, Math. Nachr. 30 (1965), 47-61. MR 0197804 (33:5967)
  • [3] W. A. Al-Salam and T. S. Chihara, Convolutions of orthogonal polynomials, SIAM J. Math. Anal. 7 (1976), 16-28. MR 0399537 (53:3381)
  • [4] W. A. Al-Salam and M. E. H. Ismail, $ q$-beta integrals and the $ q$-Hermite polynomials, Pacific J. Math. 135 (1988), 209-221. MR 968609 (90c:33001)
  • [5] -, A $ q$-beta integral on the unit circle and some biorthogonal rational functions, Proc. Amer. Math. Soc. 121 (1994), 553-561. MR 1215197 (94h:33011)
  • [6] W. A. Al-Salam and A. Verma, $ q$-analogs of some biorthogonal functions, Canad. Math. Bull. 26 (1983), 225-227. MR 697805 (84e:33010)
  • [7] W. Allaway, Some properties of the $ q$-Hermite polynomials, Canad. J. Math. 32 (1980), 686-694. MR 586985 (81j:33010)
  • [8] G. E. Andrews, $ q$-series: their development and application in analysis, number theory, combinatorics, physics, and computer algebra, CBMS Regional Conf. Ser. in Math., vol. 66, Amer. Math. Soc., Providence, RI, 1986. MR 858826 (88b:11063)
  • [9] G. E. Andrews and R. A. Askey, Classical orthogonal polynomials, Polynomes Orthogonaux et Application (C. Breziniski et al., eds.), Lecture Notes in Math., vol. 1171, Springer-Verlag, Berlin, 1984, pp. 36-63. MR 838970 (88c:33015b)
  • [10] R. A. Askey, Beta integrals and $ q$-extensions, Proc. Ramanujan Centenial Internat. Conf., Annamalainagar (R. Balakrishnan et al., eds.), Ramanujan Math. Soc., Annamalai University, 1988, pp. 85-102.
  • [11] -, Beta integrals and associated orthogonal polynomials, Number Theory, Madras 1987 (K. Alladi, ed.), Lecture Notes in Math., vol. 1395, Springer-Verlag, New York, 1989, pp. 84-121. MR 1019328 (90k:33001)
  • [12] -, Continuous $ q$-Hermite polynomials when $ q > 1$, $ q$-Series and Partitions, (D. Stanton, ed)., IMA Vol. Math. Appl., Springer-Verlag, New York, 1989, 151-158.
  • [13] R. A. Askey and M. E. H. Ismail, The very well poised $ {}_6{\psi _6}$, Proc. Amer. Math. Soc. 77 (1979), 218-222. MR 542088 (80m:33002)
  • [14] -, The Rogers $ q$-ultraspherical polynomials, Approximation Theory III (W. Cheny, ed.), Academic Press, New York, 1980, pp. 175-182.
  • [15] -, A generalization of ultraspherical polynomials, Studies in Pure Mathematics (P. Erdös, ed.), Birkhäuser, Basel, 1983, pp. 55-78. MR 820210 (87a:33015)
  • [16] -, Recurrence relations, continued fractions and orthogonal polynomials, Mem. Amer. Math. Soc. 300 (1984). MR 743545 (85g:33008)
  • [17] R. A. Askey and J. A. Wilson, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc. 319 (1985). MR 783216 (87a:05023)
  • [18] C. Berg and G. Valent, The Nevanlinna parameterization for some indeterminate Stieltjes moment problems associated with birth and death processes, Methods and Applications of Analysis (to appear). MR 1291292 (95i:44009)
  • [19] C. Brezinski, Biorthogonality and its applications to numerical analysis, Marcel Dekker, New York, 1992. MR 1151616 (92m:41001)
  • [20] L. Carlitz, Generating functions for certain $ q$-orthogonal polynomials, Collectanea Math. 23 (1972), 91-104. MR 0316773 (47:5321)
  • [21] T. S. Chihara, An introduction to orthogonal polynomials, Gordon and Breach, New York, 1978. MR 0481884 (58:1979)
  • [22] T. S. Chihara and M. E. H. Ismail, Extremal measures for a system of orthogonal polynomials, Constr. Approx. 9 (1993), 111-119. MR 1198526 (94b:33020)
  • [23] D. Foata, Some Hermite polynomial identities and their combinatorics, Adv. Appl. Math. 2 (1981), 250-259. MR 626861 (84a:33021)
  • [24] G. Freud, Orthogonal polynomials, English transl., Pergamon Press, New York, 1971.
  • [25] G. Gasper and M. Rahman, Basic hypergeometric series, Cambridge Univ. Press, Cambridge, 1990. MR 1052153 (91d:33034)
  • [26] U. Grenander and G. Szegő, Toeplitz forms and their applications, Univ. of California Press, Berkely, 1958; reprinted by Chelsea, New York, 1984. MR 0094840 (20:1349)
  • [27] R. A. Gustafson, A generalization of Selberg's beta integral, Bull. Amer. Math. Soc. (N.S.) 22 (1990), 97-108. MR 1001607 (90j:33001)
  • [28] M. E. H. Ismail, A queueing model and a set of orthogonal polynomials J. Math. Anal. Appl. 108 (1985), 575-594. MR 793667 (86k:60158)
  • [29] M. E. H. Ismail and D. R. Masson, Extremal measures for $ q$-Hermite polynomials when $ q > 1$, C. R. Math. Rep. Acad. Sci. Canada 15 (1993), 7-12. MR 1214209 (94i:33024)
  • [30] M. E. H. Ismail and D. Stanton, On the Askey-Wilson and Rogers polynomials, Canad. J. Math. 40 (1988), 1025-1045. MR 973507 (89m:33003)
  • [31] M. E. H. Ismail, D. Stanton, and G. Viennot, The combinatorics of the $ q$-Hermite polynomials and the Askey-Wilson integral, European J. Combin. 8 (1987), 379-392. MR 930175 (89h:33015)
  • [32] M. E. H. Ismail and J. Wilson, Asymptotic and generating relations for the $ q$-Jacobi and the $ {}_4{\phi _3}$ polynomials, J. Approx. Theory 36 (1982), 43-54. MR 673855 (84e:33012)
  • [33] W. F. Kibble, An extension of theorem of Mehler on Hermite polynomials, Proc. Cambridge, Philos. Soc. 41 (1945), 12-15. MR 0012728 (7:65f)
  • [34] G. Labelle, Tableau d'Askey, Polynomes Orthogonaux et Applications, (C. Breziniski et al., eds.), Lecture Notes in Math., vol. 1171, Springer-Verlag, Berlin, 1984, pp. 36-37.
  • [35] J. E. Littlewood, On the asymptotic approximation to integral functions of zero order, Proc. London Math. Soc. (2) 5 (1907), 361-410; reprinted in Collected Papers of J. E. Littlewood, vol. 2, Oxford Univ. Press, Oxford, 1986, pp. 1059-1109.
  • [36] J. D. Louck, Extension of the Kibble-Slepian formula for Hermite polynomials using Boson operator methods, Adv. in Appl. Math. 2 (1981), 239-249. MR 626860 (83a:81036)
  • [37] D. Moak, The $ q$-analogue of the Laguerre polynomials, J. Math. Anal. Appl. 81 (1981), 20-47. MR 618759 (82i:33018)
  • [38] P. Nevai, Orthogonal polynomials, Mem. Amer. Math. Soc. 13 (1979). MR 519926 (80k:42025)
  • [39] F. W. J. Olver, Asymptotics and special functions, Academic Press, New York, 1974. MR 0435697 (55:8655)
  • [40] H. Rademacher, Topics in analytic number theory, Springer-Verlag, New York, 1973. MR 0364103 (51:358)
  • [41] M. Rahman, An integral representation of a $ _{10}{\phi _9}$ and continuous biorthogonal $ _{10}{\phi _9}$ rational functions, Canad. J. Math. 38 (1986), 605-618. MR 845667 (87i:33011)
  • [42] -, Biorthogonality of a system of rational functions with respect to a positive measure on $ [ - 1,1]$, SIAM J. Math. Anal. 22 (1991), 1421-1431. MR 1112517 (92h:33016)
  • [43] -, Private communication.
  • [44] L. J. Rogers, On the expansion of some infinite products, Proc. London Math. Soc. 24 (1893), 337-352.
  • [45] -, Second memoir on the expansion of certain infinite products, Proc. London Math. Soc. 25 (1894), 318-343.
  • [46] -, Third memoir on the expansion of certain infinite products, Proc. London Math. Soc. 26 (1895), 15-32.
  • [47] J. Shohat and J. D. Tamarkin, The problem of moments, rev. ed., Amer. Math. Soc., Providence, RI, 1950. MR 0008438 (5:5c)
  • [48] D. Slepian, On the symmetrized Kronecker power of a matrix and extensions of Mehler's formula for Hermite polynomials, SIAM J. Math. Anal. 3 (1972), 606-616. MR 0315173 (47:3722)
  • [49] M. H. Stone, Linear transformations in Hilbert space and their application to analysis, Amer. Math. Soc., Providence, RI, 1932. MR 1451877 (99k:47001)
  • [50] G. Szegő, Orthogonal polynomials, 4th ed., Amer. Math. Soc., Providence, RI, 1975.
  • [51] M. Toda, Theory of nonlinear lattices, 2nd enlarged ed., Springer-Verlag, Berlin, 1988. MR 618652 (82k:58052b)
  • [52] E. T. Whittaker and G. N. Watson, A course of modern analysis, 2nd ed., Cambridge Univ. Press, Cambridge, 1927. MR 1424469 (97k:01072)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1994-1264148-6
Keywords: Hamburger moment problem, indeterminate moment problem, Bailey's sum, Poisson kernel, biorthogonal rational functions, $ q$-beta integrals, asymptotics, $ q$-Hermite polynomials, theta functions, generating functions, Askey-Wilson integral
Article copyright: © Copyright 1994 American Mathematical Society

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