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Jacobi polynomial expansions of Jacobi polynomials with non-negative coefficients

Published online by Cambridge University Press:  24 October 2008

Richard Askey  and
George Gasper
Richard Askey
Affiliation:
Supported in part by a fellowship from the John Simon Guggenheim Memorial Foundation, in part by the Office of Naval Research under contract N00014-67-A-0128-0012 and in part by the Mathematisch Centrum, Amsterdam.
George Gasper
Affiliation:
Supported in part by the National Research Council of Canada under Grant No. A-4048.
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Extract

The answers to many important questions in the harmonic analysis of orthogonal polynomials are known to depend on the determination of when formulas of the types

and their duals

hold, where pn(x) and qn(x) are suitably normalized orthogonal polynomials or orthogonal polynomials multiplied by certain functions; e.g. e−pxLn(x).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 70 , Issue 2 , September 1971 , pp. 243 - 255
Copyright
Copyright © Cambridge Philosophical Society 1971

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References

REFERENCES

(1)Askey, R.Orthogonal expansions with positive coefficients. Proc. Amer. Math. Soc. 16 (1965), 1191–1194.CrossRefGoogle Scholar
(2)Askey, R.Jacobi polynomial expansions with positive coefficients and imbeddings of projective spaces. Bull. Amer. Math. Soc. 74 (1968), 301–304.CrossRefGoogle Scholar
(3)Askey, R.Orthogonal expansions with positive coefficients. II. SIAM J. Math. Anal. 2 (1971).CrossRefGoogle Scholar
(4)Askey, R. and Fitch, J.Integral Representations for Jacobi polynomials and some applications. J. Math. Anal. Appl. 26 (1969), 411–437.CrossRefGoogle Scholar
(5)Askey, R. and Wainger, S.A transplantation theorem for ultraspherical coefficients. Pacific J. Math. 16 (1966), 393–405.CrossRefGoogle Scholar
(6)Bailey, W. W.Generalized hypergeometric series. Cambridge University Press, New York (1935).Google Scholar
(7)Bailey, W. W.Contiguous hypergeometric functions of the type 3F 2(1). Proc. Glasgow Math. Assoc. 2 (1954), 62–65.CrossRefGoogle Scholar
(8)Erdélyi, A.Higher transcendental functions, vol. 2 (McGraw-Hill, New York, 1953).Google Scholar
(9)Feldheim, E.Contributions a la theorie des polynomes do Jacobi. Mat. Fir. Lapok 48 (1941), 453–504 (Hungarian, French summary).Google Scholar
(10)Fields, J. L.Asymptotic expansions of a class of hypergeometric polynomials with respect to the order, III. J. Math. Anal. Appi. 12 (1965), 593–601.CrossRefGoogle Scholar
(11)Gasper, G.Linearization of the product of Jacobi polynomials I. Canad. J. Math. 22 (1970), 171–175.CrossRefGoogle Scholar
(12)Gasper, G.Linearization of the product of Jacobi polynomials, II. Canad. J. Math. 22 (1970), 582–593.CrossRefGoogle Scholar
(13)Gasper, G.Positivity and the convolution structure for Jacobi series. Ann. of Math. 93 (1971), 112–118.CrossRefGoogle Scholar
(14)Hahn, W. Über Orthogonalpolynome, die q-Differenzengleichungen genügen. Math. Nachr. 2 (1949), 4–34.CrossRefGoogle Scholar
(15)Kingman, J. F. C.Random walks with spherical symmetry. Acta Math. 109 (1963), 11–53.CrossRefGoogle Scholar
(16)Miller, W. Jr., Special functions and the complex Euclidean group in 3-space, II. J. Math. and Phys. 9 (1968), 1175–87.CrossRefGoogle Scholar
(17)Rainville, E. D.The contiguous function relations for DF C with applications to Bateman's and Rico's H n(ν, p, ν). Bull. Amer. Math. Soc. 51 (1945), 714–723.CrossRefGoogle Scholar
(18)Rivlin, T. J. and Wilson, M. W.An optimal property of Chebyshev expansions. J. Approx. Theory 2 (1969), 312–317.CrossRefGoogle Scholar
(19)Schoenberg, I. J.Metric spaces and completely monotone functions. Ann. of Math. 39 (1938), 811–841.CrossRefGoogle Scholar
(20)Szegö, G.Über gewisse Potenzrethen mit lauter positiven Koeffizienten. Math. Z. 37 (1933), 674–688.CrossRefGoogle Scholar
(21)Szegö, G.Orthogonal Polynomials. Amer. Math. Soc. Colloq. Pub., vol. 23, Amer. Math. Soc., Providence R. I. (1967).Google Scholar
(22)Wilson, M. W.A new discrete analog of the Legenthe Polynomials. Bull. Amer. Math. Soc. 76 (1970), 345–348.CrossRefGoogle Scholar
(23)Wilson, M. W.Non-negative expansions of polynomials. Proc. Amer. Math. Soc. 24 (1970), 100–102.CrossRefGoogle Scholar