Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A projection formula for the Askey-Wilson polynomials and an application


Author: Mizan Rahman
Journal: Proc. Amer. Math. Soc. 103 (1988), 1099-1107
MSC: Primary 33A65; Secondary 05A30
MathSciNet review: 954990
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Abstract: A projection formula for $ {p_n}(x;a,b,c,d\vert q)$, the Askey-Wilson polynomials, is obtained by using a generalization of Askey and Wilson's $ q$-beta integral. The result is used to find a $ q$-analogue of the Feldheim-Vilenkin formula for ultraspherical polynomials. A $ q$-analogue of the ultraspherical polynomials, other than the one due to Rogers, is also introduced.


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  • [1] Richard Askey, Orthogonal polynomials and special functions, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1975. MR 0481145
  • [2] Richard Askey, Jacobi polynomials. I. New proofs of Koornwinder’s Laplace type integral representation and Bateman’s bilinear sum, SIAM J. Math. Anal. 5 (1974), 119–124. MR 0385197
  • [3] Richard Askey and James Fitch, Integral representations for Jacobi polynomials and some applications., J. Math. Anal. Appl. 26 (1969), 411–437. MR 0237847
  • [4] R. Askey and Mourad E. H. Ismail, A generalization of ultraspherical polynomials, Studies in pure mathematics, Birkhäuser, Basel, 1983, pp. 55–78. MR 820210
  • [5] -, The Rogers $ q$-ultraspherical polynomials, Approximation Theory. III (E. W. Cheney, ed.), Academic Press, New York, 1980, pp. 175-182.
  • [6] 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, 10.1090/memo/0319
  • [7] W. N. Bailey, Generalized hypergeometric series, Cambridge Tracts in Mathematics and Mathematical Physics, No. 32, Stechert-Hafner, Inc., New York, 1964. MR 0185155
  • [8] A. Erdélyi, et al., Higher transcendental functions, vol. 2, McGraw-Hill, New York, 1953.
  • [9] L. Fejér, Sur le développement d'une fonction arbitraire suivant les fonctions de Laplace, C. R. Acad. Sci. Paris 146 (1908), 224-225; reproduced in Gesammelte Arbeiten I. 319-322.
  • [10] Ervin Feldheim, On the positivity of certain sums of ultraspherical polynomials, J. Analyse Math. 11 (1963), 275–284. MR 0158107
  • [11] George Gasper and Mizan Rahman, Positivity of the Poisson kernel for the continuous 𝑞-Jacobi polynomials and some quadratic transformation formulas for basic hypergeometric series, SIAM J. Math. Anal. 17 (1986), no. 4, 970–999. MR 846401, 10.1137/0517069
  • [12] T. H. Koornwinder, The addition formula for Jacobi polynomials. I. Summary of results, Nederl. Akad. Wetensch. Proc. Ser. A 75=Indag. Math. 34 (1972), 188–191. MR 0308476
  • [13] Tom Koornwinder, Jacobi polynomials. II. An analytic proof of the product formula, SIAM J. Math. Anal. 5 (1974), 125–137. MR 0385198
  • [14] Tom Koornwinder, Jacobi polynomials. III. An analytic proof of the addition formula, SIAM. J. Math. Anal. 6 (1975), 533–543. MR 0447659
  • [15] Thomas P. Laine, Projection formulas and a new proof of the addition formula for the Jacobi polynomials, SIAM J. Math. Anal. 13 (1982), no. 2, 324–330. MR 647130, 10.1137/0513024
  • [16] B. Nassrallah and Mizan Rahman, Projection formulas, a reproducing kernel and a generating function for 𝑞-Wilson polynomials, SIAM J. Math. Anal. 16 (1985), no. 1, 186–197. MR 772878, 10.1137/0516014
  • [17] Mizan Rahman, An integral representation of a ₁₀𝜙₉ and continuous bi-orthogonal ₁₀𝜙₉ rational functions, Canad. J. Math. 38 (1986), no. 3, 605–618. MR 845667, 10.4153/CJM-1986-030-6
  • [18] Mizan Rahman, The linearization of the product of continuous 𝑞-Jacobi polynomials, Canad. J. Math. 33 (1981), no. 4, 961–987. MR 634153, 10.4153/CJM-1981-076-8
  • [19] Mizan Rahman, A product formula for the continuous 𝑞-Jacobi polynomials, J. Math. Anal. Appl. 118 (1986), no. 2, 309–322. MR 852163, 10.1016/0022-247X(86)90265-9
  • [20] D. B. Sears, On the transformation theory of basic hypergeometric functions, Proc. London Math. Soc. (2) 53 (1951), 158–180. MR 0041981
  • [21] Lucy Joan Slater, Generalized hypergeometric functions, Cambridge University Press, Cambridge, 1966. MR 0201688
  • [22] Gábor Szegő, Orthogonal polynomials, 4th ed., American Mathematical Society, Providence, R.I., 1975. American Mathematical Society, Colloquium Publications, Vol. XXIII. MR 0372517
  • [23] N. Ya. Vilenkin, Some relations for Gegenbauer functions, Uspehi Mat. Nauk (N.S.) 13 (1958), no. 3(81), 167–172 (Russian). MR 0095986

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1988-0954990-3
Keywords: $ q$-ultraspherical and Askey-Wilson polynomials, Feldheim-Vilenkin formula, balanced and very-well-poised basic hypergeometric series
Article copyright: © Copyright 1988 American Mathematical Society