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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Sieved ultraspherical polynomials
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by Waleed Al-Salam, W. R. Allaway and Richard Askey PDF
Trans. Amer. Math. Soc. 284 (1984), 39-55 Request permission

Abstract:

The continuous $q$-ultraspherical polynomials contain a number of important examples as limiting or special cases. One of these arose in Allaway’s Ph.D. thesis. In a previous paper we solved a characterization problem essentially equivalent to Allaway’s and showed that these polynomials arose from the $q$-ultraspherical polynomials when $q$ approached a root of unity. A second class of such polynomials is found, and the recurrence relation and orthogonality relation are found for each of these polynomials. The orthogonality is interesting because the weight function has a finite number of zeros in $(-1, 1)$. Generating functions and other formulas are also found.
References
  • Waleed Al-Salam, Wm. R. Allaway, and Richard Askey, A characterization of the continuous $q$-ultraspherical polynomials, Canad. Math. Bull. 27 (1984), no. 3, 329–336. MR 749641, DOI 10.4153/CMB-1984-050-x
    • Wm. R. Allaway, The identification of a class of orthogonal polynomials, Ph.D. thesis, University of Alberta, Canada, 1972.
  • George E. Andrews, The theory of partitions, Encyclopedia of Mathematics and its Applications, Vol. 2, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. MR 0557013
  • Richard Askey, The $q$-gamma and $q$-beta functions, Applicable Anal. 8 (1978/79), no. 2, 125–141. MR 523950, DOI 10.1080/00036817808839221
  • Richard Askey, Ramanujan’s extensions of the gamma and beta functions, Amer. Math. Monthly 87 (1980), no. 5, 346–359. MR 567718, DOI 10.2307/2321202
  • Richard A. Askey and Mourad E. H. Ismail, The Rogers $q$-ultraspherical polynomials, Approximation theory, III (Proc. Conf., Univ. Texas, Austin, Tex., 1980), Academic Press, New York-London, 1980, pp. 175–182. MR 602713
  • R. Askey and Mourad E. H. Ismail, A generalization of ultraspherical polynomials, Studies in pure mathematics, Birkhäuser, Basel, 1983, pp. 55–78. MR 820210
    • —, Recurrence relations, continued fractions and orthogonal polynomials (to appear).
  • 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, DOI 10.1090/memo/0319
  • F. V. Atkinson and W. N. Everitt, Orthogonal polynomials which satisfy second order differential equations, E. B. Christoffel (Aachen/Monschau, 1979) Birkhäuser, Basel-Boston, Mass., 1981, pp. 173–181. MR 661063
    • W. N. Bailey, The generating function of Jacobi polynomials, J. London Math. Soc. 13 (1938), 8-12. —, Generalized hypergeometric series, Hafner, New York, 1972.
  • D. M. Bressoud, Theta function identities and orthogonal polynomials, Analytic number theory (Philadelphia, Pa., 1980) Lecture Notes in Math., vol. 899, Springer, Berlin-New York, 1981, pp. 325–332. MR 654538
  • T. S. Chihara, An introduction to orthogonal polynomials, Mathematics and its Applications, Vol. 13, Gordon and Breach Science Publishers, New York-London-Paris, 1978. MR 0481884
  • E. Feldheim, Sur les polynomes généralisés de Legendre, Bull. Acad. Sci. URSS. Sér. Math. [Izvestia Akad. Nauk SSSR] 5 (1941), 241–254 (French., with Russian translation summary). MR 0005173
  • George Gasper and Mizan Rahman, Positivity of the Poisson kernel for the continuous $q$-ultraspherical polynomials, SIAM J. Math. Anal. 14 (1983), no. 2, 409–420. MR 688587, DOI 10.1137/0514034
  • I. L. Lanzewizky, Über die Orthogonalität der Féjèr-Szegöschen Polynome, C. R. (Doklady) Acad. Sci. URSS (N. S.) 31 (1941), 199–200 (German). MR 0005172
  • Paul G. Nevai, Orthogonal polynomials defined by a recurrence relation, Trans. Amer. Math. Soc. 250 (1979), 369–384. MR 530062, DOI 10.1090/S0002-9947-1979-0530062-6
    • L. J. Rogers, Third memoir on the expansion of certain infinite products, Proc. London Math. Soc. 26 (1895), 15-32.
  • Lucy Joan Slater, Generalized hypergeometric functions, Cambridge University Press, Cambridge, 1966. MR 0201688
    • G. Szegö, Orthogonal polynomials, 4th ed., Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, R.I., 1975.
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Additional Information
  • © Copyright 1984 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 284 (1984), 39-55
  • MSC: Primary 33A45; Secondary 33A65, 42C05
  • DOI: https://doi.org/10.1090/S0002-9947-1984-0742411-6
  • MathSciNet review: 742411