Geodesic
A Non-Negative Representation of the Linearization Coefficients of the Product of Jacobi Polynomials
Canadian journal of mathematics, Tome 33 (1981) no. 4, pp. 915-928

Voir la notice de l'article provenant de la source Cambridge University Press

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) 
Rahman, Mizan. A Non-Negative Representation of the Linearization Coefficients of the Product of Jacobi Polynomials. Canadian journal of mathematics, Tome 33 (1981) no. 4, pp. 915-928. doi: 10.4153/CJM-1981-072-9
@article{10_4153_CJM_1981_072_9,
     author = {Rahman, Mizan},
     title = {A {Non-Negative} {Representation} of the {Linearization} {Coefficients} of the {Product} of {Jacobi} {Polynomials}},
     journal = {Canadian journal of mathematics},
     pages = {915--928},
     year = {1981},
     volume = {33},
     number = {4},
     doi = {10.4153/CJM-1981-072-9},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1981-072-9/}
}
TY  - JOUR
AU  - Rahman, Mizan
TI  - A Non-Negative Representation of the Linearization Coefficients of the Product of Jacobi Polynomials
JO  - Canadian journal of mathematics
PY  - 1981
SP  - 915
EP  - 928
VL  - 33
IS  - 4
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-1981-072-9/
DO  - 10.4153/CJM-1981-072-9
ID  - 10_4153_CJM_1981_072_9
ER  - 
%0 Journal Article
%A Rahman, Mizan
%T A Non-Negative Representation of the Linearization Coefficients of the Product of Jacobi Polynomials
%J Canadian journal of mathematics
%D 1981
%P 915-928
%V 33
%N 4
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-1981-072-9/
%R 10.4153/CJM-1981-072-9
%F 10_4153_CJM_1981_072_9

[1] 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] 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] 3. Askey, R. and Wainger, S., Orthogonal polynomials and special functions, Regional conference series in Applied Mathematics & f (SIAM, Philadelphia, 1975). Google Scholar

[4] 4. Bailey, W. N., Generalized hyper geometric series (Stechert-Hafner Service Agency, New York and London, 1964). Google Scholar

[5] 5. Carlitz, L., The product of two ultraspherical polynomials, Proc. Glasgow Math. Assoc. 5 (1961/2), 76–79. Google Scholar

[6] 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] 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] 8. Gasper, G., Linearization of the product of Jacobi polynomials, I, Can. J. Math. 22 (1970), 171–175. Google Scholar

[9] 9. Gasper, G., Linearization of the product of Jacobi polynomials, II, Can. J. Math. 22 (1970), 582–593. Google Scholar

[10] 10. Gasper, G., Projection formulas for orthogonal polynomials of a discrete variable, J. Math. Anal. Appl. 45 (1974), 176–198. Google Scholar

[11] 11. Gasper, G., Positivity and special functions, in Theory and application of special functions (Academic Press, New York, 1975), 375–433. Google Scholar

[12] 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] 13. Hylleraas, E., Linearization of products of Jacobi polynomials, Math. Scand. 10 (1962), 189–200. Google Scholar

[14] 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] 15. Miller, W., Special functions and the complex Euclidean group in 3-space, I., J. Math. Phys. 9 (1968), 1163–1175. Google Scholar

[16] 16. Miller, W., Special functions and the complex Euclidean group in 3-space, IT, J. Math. Phys. 9 (1968), 1175–1187. Google Scholar

[17] 17. Slater, L. J., Generalized hyper geometric functions (Cambridge University Press, 1966). Google Scholar

[18] 18. Szegö, G., Orthogonal polynomials, Amer. Math. Soc. Coll. Publications, Vol. 23, 4th ed. (1975). Google Scholar

Cité par Sources :