Abstract
Aq-integral representation of Rogers'q-ultraspherical polynomialsC n (x;β∥q) is obtained by using Sears' summation formula for balanced non-terminating3 φ 2 series. It is then used to give a simple derivation of the Gasper-Rahman formula for the Poisson kernel ofC n (x;β∥q). As another application it is shown how this representation can be directly used to give an asymptotic expansion of theq-ultraspherical polynomials.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
W. A. Al-Salam, A. Verma (1982):Some remarks on q-beta integral. Proc. Amer. Math. Soc.,85:360–362.
R. Askey, M. E. H. Ismail (1980):The Rogers' q-ultraspherical polynomials. In: Approximation Theory III (E. W. Cheney, ed.). New York: Academic Press, pp. 175–182.
R. Askey, M. E. H. Ismail (1982):A generalization of ultraspherical polynomials. In: Studies in Pure Mathematics (P. Erdös, ed.). Boston: Birkhäuser, pp. 56–78.
W. N. Bailey (1964):Generalized Hypergeometric Series. New York, London: Stechert-Hafner Service Agency.
D. M. Bressoud (1981):Linearization and related formulas for q-ultraspherical polynomials. SIAM J. Math. Anal.,12:161–168.
G. Gasper (1985):Rogers' linearization formula for the continuous q-ultraspherical polynomials and quadratic transformation formulas. SIAM J. Math. Anal.16:1061–1071.
G. Gasper, M. Rahman (1983):Positivity of the Poisson kernel for the continuous q-ultraspherical polynomials. SIAM J. Math. Anal.,14:409–420.
G. Gasper, M. Rahman (in press):Positivity of the Poisson kernel for the continuous q-Jacobi polynomials and some quadratic transformation formulas for basic hypergeometric series.
E. Heine (1847):Untersuchungen über die Reihe⋯. J. Reine Angew. Math.,34:285–328.
M. Rahman (1981):The linearization of the product of continuous q-Jacobi polynomials. Canad. J. Math.,33:961–987.
L. J. Rogers (1895):Third memoir on the expansion of certain infinite products. Proc. London Math. Soc.,26:15–32.
D. B. Sears (1951):Transformations of basic hypergeometric functions of special type. Proc. London Math. Soc., (1),52:467–483.
D. B. Sears (1951):On the transformation theory of basic hypergeometric functions. Proc. London Math. Soc. (2),53:158–191.
G. Szegö (1934):Über gewisse orthogonale Polynome, die zu einer oszillierenden Belegunsfunktion gehören. Math. Ann.,110:501–513.
G. Szegö (1975): Orthogonal Polynomials, 4th ed., Vol. 23. Providence: Colloquium Publications, American Mathematical Society.
G. Szegö (1982): Collected Papers, Vol. 2. Boston: Birkhäuser, pp. 545–557.
Author information
Authors and Affiliations
Additional information
Communicated by Edward B. Saff.
Rights and permissions
About this article
Cite this article
Rahman, M., Verma, A. A q-integral representation of Rogers' q-ultraspherical polynomials and some applications. Constr. Approx 2, 1–10 (1986). https://doi.org/10.1007/BF01893413
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01893413