Sieved ultraspherical polynomials

Authors:
Waleed Al-Salam, W. R. Allaway and Richard Askey

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

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Abstract: The continuous -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 -ultraspherical polynomials when 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 . Generating functions and other formulas are also found.

**[1]**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**, https://doi.org/10.4153/CMB-1984-050-x**[2]**Wm. R. Allaway,*The identification of a class of orthogonal polynomials*, Ph.D. thesis, University of Alberta, Canada, 1972.**[3]**George E. Andrews,*The theory of partitions*, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. Encyclopedia of Mathematics and its Applications, Vol. 2. MR**0557013****[4]**Richard Askey,*The 𝑞-gamma and 𝑞-beta functions*, Applicable Anal.**8**(1978/79), no. 2, 125–141. MR**523950**, https://doi.org/10.1080/00036817808839221**[5]**Richard Askey,*Ramanujan’s extensions of the gamma and beta functions*, Amer. Math. Monthly**87**(1980), no. 5, 346–359. MR**567718**, https://doi.org/10.2307/2321202**[6]**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****[7]**R. Askey and Mourad E. H. Ismail,*A generalization of ultraspherical polynomials*, Studies in pure mathematics, Birkhäuser, Basel, 1983, pp. 55–78. MR**820210****[8]**-,*Recurrence relations, continued fractions and orthogonal polynomials*(to appear).**[9]**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**[10]**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****[11]**W. N. Bailey,*The generating function of Jacobi polynomials*, J. London Math. Soc.**13**(1938), 8-12.**[12]**-,*Generalized hypergeometric series*, Hafner, New York, 1972.**[13]**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****[14]**T. S. Chihara,*An introduction to orthogonal polynomials*, Gordon and Breach Science Publishers, New York-London-Paris, 1978. Mathematics and its Applications, Vol. 13. MR**0481884****[15]**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****[16]**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**, https://doi.org/10.1137/0514034**[17]**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****[18]**Paul G. Nevai,*Orthogonal polynomials defined by a recurrence relation*, Trans. Amer. Math. Soc.**250**(1979), 369–384. MR**530062**, https://doi.org/10.1090/S0002-9947-1979-0530062-6**[19]**L. J. Rogers,*Third memoir on the expansion of certain infinite products*, Proc. London Math. Soc.**26**(1895), 15-32.**[20]**Lucy Joan Slater,*Generalized hypergeometric functions*, Cambridge University Press, Cambridge, 1966. MR**0201688****[21]**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

DOI:
https://doi.org/10.1090/S0002-9947-1984-0742411-6

Keywords:
Sieved ultraspherical polynomials of first and second kind,
continuous -ultraspherical polynomials,
Fejér-Legendre polynomials,
orthogonal polynomials,
ultraspherical polynomials,
Tchebycheff polynomials,
recurrence relation,
weight function,
generating function,
product linearization,
connection coefficient

Article copyright:
© Copyright 1984
American Mathematical Society