A right inverse of the Askey-Wilson operator

Authors:
B. Malcolm Brown and Mourad E. H. Ismail

Journal:
Proc. Amer. Math. Soc. **123** (1995), 2071-2079

MSC:
Primary 33D20; Secondary 33D45, 39A70, 42C10, 45E10

DOI:
https://doi.org/10.1090/S0002-9939-1995-1273478-X

MathSciNet review:
1273478

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Abstract | References | Similar Articles | Additional Information

Abstract: We establish an integral representation of a right inverse of the Askey-Wilson finite difference operator on with weight . The kernel of this integral operator is and is the Riemann mapping function that maps the interior of an ellipse conformally onto the open unit disc.

**[1]**R. Askey and Mourad E. H. Ismail,*A generalization of ultraspherical polynomials*, Studies in pure mathematics, Birkhäuser, Basel, 1983, pp. 55–78. MR**820210****[2]**Richard Askey and James Wilson,*Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials*, Mem. Amer. Math. Soc.**54**(1985), no. 319, iv+55. MR**783216**, https://doi.org/10.1090/memo/0319**[3]**George Gasper and Mizan Rahman,*Basic hypergeometric series*, Encyclopedia of Mathematics and its Applications, vol. 35, Cambridge University Press, Cambridge, 1990. With a foreword by Richard Askey. MR**1052153****[4]**Mourad E. H. Ismail,*The zeros of basic Bessel functions, the functions 𝐽_{𝜈+𝑎𝑥}(𝑥), and associated orthogonal polynomials*, J. Math. Anal. Appl.**86**(1982), no. 1, 1–19. MR**649849**, https://doi.org/10.1016/0022-247X(82)90248-7**[5]**Mourad E. H. Ismail and Ruiming Zhang,*Diagonalization of certain integral operators*, Adv. Math.**109**(1994), no. 1, 1–33. MR**1302754**, https://doi.org/10.1006/aima.1994.1077**[6]**Mourad E. H. Ismail, Mizan Rahman, and Ruiming Zhang,*Diagonalization of certain integral operators. II*, J. Comput. Appl. Math.**68**(1996), no. 1-2, 163–196. MR**1418757**, https://doi.org/10.1016/0377-0427(95)00263-4**[7]**Alphonse P. Magnus,*Associated Askey-Wilson polynomials as Laguerre-Hahn orthogonal polynomials*, Orthogonal polynomials and their applications (Segovia, 1986) Lecture Notes in Math., vol. 1329, Springer, Berlin, 1988, pp. 261–278. MR**973434**, https://doi.org/10.1007/BFb0083366**[8]**Zeev Nehari,*Conformal mapping*, McGraw-Hill Book Co., Inc., New York, Toronto, London, 1952. MR**0045823****[9]**Zeev Nehari,*Introduction to complex analysis*, Allyn and Bacon, Inc., Boston, Mass., 1961. MR**0224779****[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*, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions; Reprint of the fourth (1927) edition. MR**1424469**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1995-1273478-X

Keywords:
Integral operator,
Chebyshev polynomials,
theta functions,
finite difference operators,
conformal mappings,
*q*-Hermite polynomials

Article copyright:
© Copyright 1995
American Mathematical Society