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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
DOI: https://doi.org/10.1090/S0002-9939-1988-0954990-3
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] R. Askey, Orthogonal polynomials and special functions, Regional Conf. Series in Appl. Math., SIAM, Philadelphia, Pa., 1975. MR 0481145 (58:1288)
  • [2] -, Jacobi polynomials I. New proofs of Koornwinder's Laplace-type integral representations and Bateman's bilinear sum, SIAM J. Math. Anal. 5 (1974), 119-124. MR 0385197 (52:6062)
  • [3] R. Askey and J. Fitch, Integral representations for Jacobi polynomials and some applications, J. Math. Anal. Appl. 26 (1969), 411-437. MR 0237847 (38:6128)
  • [4] R. Askey and M. E. H. Ismail, A generalization of ultraspherical polynomials, Studies in Pure Mathematics (P. Erdös, ed.), Birkhauser, Boston, Mass., 1982, pp. 56-78. MR 820210 (87a:33015)
  • [5] -, The Rogers $ q$-ultraspherical polynomials, Approximation Theory. III (E. W. Cheney, ed.), Academic Press, New York, 1980, pp. 175-182.
  • [6] R. Askey and J. Wilson, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc., no 319 (1985). MR 783216 (87a:05023)
  • [7] W. N. Bailey, Generalized hypergeometric series, Stechert-Hafner, New York and London, 1964. MR 0185155 (32:2625)
  • [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] E. Feldheim, On the positivity of certain sums of ultraspherical polynomials, J. Analyse Math. 11 (1963), 275-284. MR 0158107 (28:1333)
  • [11] G. Gasper and M. 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), 970-999. MR 846401 (87i:33036)
  • [12] T. Koornwinder, The addition formula for Jacobi polynomials. I, Summary of results, Indag. Math. 34 (1972), 293-316. MR 0308476 (46:7590)
  • [13] -, Jacobi polynomials. II, An analytic proof of the product formula, SIAM J. Math. Anal. 5 (1974), 125-137. MR 0385198 (52:6063)
  • [14] -, Jacobi polynomials. III, An analytic proof of the addition formula, SIAM J. Math. Anal. 6 (1975), 533-543. MR 0447659 (56:5969)
  • [15] T. P. Laine, Projection formulas and a new proof of the addition formula for the Jacobi polynomials, SIAM J. Math. Anal. 13 (1982), 324-330. MR 647130 (83e:33005)
  • [16] B. Nassrallah and M. Rahman, Projection formulas, a reproducing kernel and a generating function for $ q$- Wilson polynomials, SIAM J. Math. Anal. 16 (1985), 186-197. MR 772878 (87b:33009)
  • [17] M. Rahman, An integral representation of a $ _{10}{\phi _9}$ and continuous biorthogonal $ _{10}{\phi _9}$ rational functions, Canad. J. Math. 38 (1986), 605-618. MR 845667 (87i:33011)
  • [18] -, The linearization of the product of continuous $ q$-Jacobi polynomials, Canad. J. Math. 33 (1981), 961-987. MR 634153 (83i:33007b)
  • [19] -, A product formula for the continuous $ q$-Jacobi polynomials, J. Math. Anal. Appl. 118 (1986), 309-322. MR 852163 (87i:33033)
  • [20] D. B. Sears, On the transformation theory of basic hypergeometric functions, Proc. London Math. Soc. (2) 53 (1951), 158-180. MR 0041981 (13:33d)
  • [21] L. J. Slater, Generalized hypergeometric functions, Cambridge Univ. Press, 1966. MR 0201688 (34:1570)
  • [22] G. Szegö, Orthogonal polynomials (4th ed.), Amer. Math. Soc. Colloq. Publ., Vol. 23, Amer. Math. Soc., Providence, R.I., 1975. MR 0372517 (51:8724)
  • [23] N. J. Vilenkin, Some relations for Gegenbauer functions, Uspekhi Mat. Nauk (N.S.) 13 (1958), no. 3(81), 167-172 (Russian). MR 0095986 (20:2484)

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Additional Information

DOI: https://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

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