Abstract
We show that a large number of explicit formulas for Padé approximants for the ratios of basic hypergeometric functions result from an explicit expression given by Ismail and Rahman for the associated Askey-Wilson polynomials. By specializing this result and using a new transformation for basic hypergeometric series, we are able to recover a result due to Andrews, Goulden and Jackson. We also show how Padé approximants off the main diagonal can be constructed in this latter case.
This author’s work was partially supported by the National Science Foundation under grants DMS 8814026 and DMS 8912423.
This author’s work was partially supported by the National Science Foundation under grants DMS 8802381.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Andrews, G. E., and Askey, R. A., Classical orthogonal polynomials, in “Polynômes Orthogonaux et Applications”, Lecture Notes in Mathematics Vol. 1171, (Eds. C. Brezinski et. al.), Springer-Verlag, Berlin, pp. 36–62 (1985).
Andrews, G. E., Goulden, IP., Jackson, D.M., Shank’s convergence acceleration transform, Padé approximants and partitions, J. Combin. Theory Ser. A 43(1986), pp. 70–84.
Askey, R. A. and Wilson, J.A., Some basic hypergeometric orthogonal polynomials that generalize the Jacobi polynomials, No. 319, Memoirs Amer. Math. Soc., Providence (1985).
Bailey, W. N., Generalized Hypergeometric Series, Cambridge University Press, Cambridge (1935).
Baker, G. A.,Jr. and Graves-Morris, P., Padé Approximants. Part I: Basic Theory, v. 13, Encyclopedia of mathematics and its applications, Addison-Wesley, Reading, Mass. (1981).
Brezinski, C, Padé Type Approximation and General Orthogonal Polynomials, Birkhäuser, Boston (1980).
Erdèlyi, A., et al, Higher Transcendental Functions, Volumes 1, 2, 3, McGraw-Hill, New York (1953).
Gasper, G. and Rahman, M., Basic Hypergeometric Series, Cambridge University Press, Cambridge (1990).
Gupta, D.P. and Masson, D.R., Exceptional Askey-Wilson polynomials, Proc. Amer. Math. Soc, to appear.
Ismail, M.E.H., Letessier, J., Valent, G., and Wimp, J., Two families of associated Wilson polynomials, Canad. J. Math. 42(1990), pp. 659–695.
Ismail, M.E.H. and Masson, D. R., Two families of orthogonal polynomials related to Jacobi polynomials, Rocky Mountain J. Math. 21(1991), pp. 359–375.
Ismail, M.E.H and Rahman, M., The associated Askey-Wilson polynomials, to appear, Trans. Amer. Math. Soc. (1991).
Ismail, M.E.H and Wilson, J., Asymptotic and generating relations for the q-Jacobi and 4Ф3 polynomials, J. Approx. Theory 36(1982), pp. 43–54.
Lubinsky, D.S. and Saff, E.B., Convergence of Padé approximants of partial theta functions and the Rogers-Szegö polynomials, Constr. Approx. 3(1987), pp. 331–361.
Masson, D.R. Wilson polynomials and some continued fractions of Ramanujan, Rocky Mountain J. Math. 21(1991), pp. 489–499.
Wimp, J., Explicit formulas for associated Jacobi polynomials and some applications, Can. J. Math. 39(1987), pp. 983–1000.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1992 Springer-Verlag New York, Inc.
About this paper
Cite this paper
Ismail, M.E.H., Perline, R., Wimp, J. (1992). Padé Approximants for Some q-Hypergeometric Functions. In: Gonchar, A.A., Saff, E.B. (eds) Progress in Approximation Theory. Springer Series in Computational Mathematics, vol 19. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-2966-7_2
Download citation
DOI: https://doi.org/10.1007/978-1-4612-2966-7_2
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4612-7737-8
Online ISBN: 978-1-4612-2966-7
eBook Packages: Springer Book Archive