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A Non-Negative Representation of the Linearization Coefficients of the Product of Jacobi Polynomials

Published online by Cambridge University Press:  20 November 2018

Mizan Rahman
Mizan Rahman*
Affiliation:
Carleton University, Ottawa, Ontario
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The problem of linearizing products of orthogonal polynomials, in general, and of ultraspherical and Jacobi polynomials, in particular, has been studied by several authors in recent years [1, 2, 9, 10, 13-16]. Standard defining relation [7, 18] for the Jacobi polynomials is given in terms of an ordinary hypergeometric function:

with Re α > –1, Re β > –1, –1 ≦ x ≦ 1. However, for linearization problems the polynomials Rn(α,β)(x), normalized to unity at x = 1, are more convenient to use:

(1.1)

Roughly speaking, the linearization problem consists of finding the coefficients g(k, m, n; α,β) in the expansion

(1.2)

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 33 , Issue 4 , 01 August 1981 , pp. 915 - 928
Copyright
Copyright © Canadian Mathematical Society 1981

References

1. Askey, R., Linearization of the product of orthogonal polynomials, in Problems in analysis (Princeton Univ. Press, Princeton, N.J., 1970), 223–228.Google Scholar
2. Askey, R. and Wainger, S., A dual convolution structure for Jacobi polynomials, in Orthogonal expansions and their continuous analogues, Proc. Conf., Edwardsville, Illinois, (Southern Illinois University Press, Carbondale, Illinois, 1968), 25–36.Google Scholar
3. Askey, R. and Wainger, S., Orthogonal polynomials and special functions, Regional conference series in Applied Mathematics & f (SIAM, Philadelphia, 1975).Google Scholar
4. Bailey, W. N., Generalized hyper geometric series (Stechert-Hafner Service Agency, New York and London, 1964).Google Scholar
5. Carlitz, L., The product of two ultraspherical polynomials, Proc. Glasgow Math. Assoc. 5 (1961/2), 76–79.Google Scholar
6. Dougall, J., A theorem of Sonine in Bessel functions, with two extensions to spherical harmonics, Proc. Edin. Math. Soc. 37 (1919), 33–47.Google Scholar
7. Erdelyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G., eds., Higher transcendental functions, Vol. II, Bateman Manuscript Project (McGraw-Hill, New York, 1953).Google Scholar
8. Gasper, G., Linearization of the product of Jacobi polynomials, I, Can. J. Math. 22 (1970), 171–175.Google Scholar
9. Gasper, G., Linearization of the product of Jacobi polynomials, II, Can. J. Math. 22 (1970), 582–593.Google Scholar
10. Gasper, G., Projection formulas for orthogonal polynomials of a discrete variable, J. Math. Anal. Appl. 45 (1974), 176–198.Google Scholar
11. Gasper, G., Positivity and special functions, in Theory and application of special functions (Academic Press, New York, 1975), 375–433.Google Scholar
12. Hsu, H. Y., Certain integrals and infinite series involving ultraspherical polynomials and Bessel functions, Duke Math. J. 4 (1938), 374–383.Google Scholar
13. Hylleraas, E., Linearization of products of Jacobi polynomials, Math. Scand. 10 (1962), 189–200.Google Scholar
14. Koornwinder, T., Positivity proofs for linearization and connection coefficients of orthogonal polynomials satisfying an addition formula, J. Lond. Math. Soc. (2) 18 (1978), 101–114.Google Scholar
15. Miller, W., Special functions and the complex Euclidean group in 3-space, I., J. Math. Phys. 9 (1968), 1163–1175.Google Scholar
16. Miller, W., Special functions and the complex Euclidean group in 3-space, IT, J. Math. Phys. 9 (1968), 1175–1187.Google Scholar
17. Slater, L. J., Generalized hyper geometric functions (Cambridge University Press, 1966).Google Scholar
18. Szegö, G., Orthogonal polynomials, Amer. Math. Soc. Coll. Publications, Vol. 23, 4th ed. (1975).Google Scholar