Proceedings of the American Mathematical Society

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ISSN 1088-6826(e) ISSN 0002-9939(p)

A right inverse of the Askey-Wilson operator

Author(s): B. Malcolm Brown; Mourad E. H. Ismail
Journal: Proc. Amer. Math. Soc. 123 (1995), 2071-2079.
MSC: Primary 33D20; Secondary 33D45, 39A70, 42C10, 45E10
MathSciNet review: 1273478
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Abstract: We establish an integral representation of a right inverse of the Askey-Wilson finite difference operator on ${L^2}$ with weight ${(1 - {x^2})^{ - 1/2}}$ . The kernel of this integral operator is $\vartheta _4'/\vartheta_4$ and is the Riemann mapping function that maps the interior of an ellipse conformally onto the open unit disc.

References:

[1]: R. Askey and M. E. H. Ismail, A generalization of ultraspherical polynomials, Studies in Pure Mathematics (P. Erdös, ed.), Birkhauser, Basel, 1983, pp. 55-78. MR 820210 (87a:33015)
[2]: R. Askey and J. Wilson, Some basic hypergeometric polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc., Number 319, Amer. Math. Soc., Providence, RI, 1985. MR 783216 (87a:05023)
[3]: G. Gasper and M. Rahman, Basic hypergeometric series, Cambridge Univ. Press, Cambridge, 1990. MR 1052153 (91d:33034)
[4]: M. E. H. Ismail, The zeros of basic Bessel functions, the functions ${J_{v + ax}}(x)$ , and associated orthogonal polynomials, J. Math. Anal. Appl. 82 (1982), 1-19. MR 649849 (83c:33010)
[5]: M. E. H. Ismail and R. Zhang, Diagonalization of certain integral operators, Adv. Math. 109 (1994), 1-33. MR 1302754 (96d:39005)
[6]: M. E. H. Ismail, M. Rahman, and R. Zhang, Diagonalization of certain integral operators II, J. Comput. Appl. Math, (to appear). MR 1418757 (98d:33011)
[7]: A. P. Magnus, Associated Askey-Wilson polynomials as Laguerre-Hahn orthogonal polynomials, Orthogonal Polynomials and Their Applications (M. Alfaro et al., eds.), Lecture Notes in Math., vol. 1329, Springer-Verlag, Berlin, 1988, pp. 261-278. MR 973434 (90d:33008)
[8]: Z. Nehari, Conformal mapping, McGraw-Hill, New York, 1952. MR 0045823 (13:640h)
[9]: -, Introduction to complex analysis, Allyn and Bacon, Boston, MA, 1961. MR 0224779 (37:378)
[10]: G. Szegö, Orthogonal polynomials, fourth edition, Amer. Math. Soc., Providence, RI, 1975.
[11]: E. T. Whittaker and G. N. Watson, A course of modern analysis, second edition, Cambridge Univ. Press, Cambridge, 1927. MR 1424469 (97k:01072)

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