Geodesic
Linearization of the Product of Jacobi Polynomials. III
Canadian journal of mathematics, Tome 23 (1971) no. 2, pp. 332-338

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In a series of papers [1; 2; 3; 4] the operation of linearizing the product of two Jacobi polynomials Pn (α, β)(x), α, β > –1, has been investigated and the existence of a natural Banach algebra associated with the linearization coefficients has been proven. This was proven for α+ β + 1 ≧ 0 in [3] and for a slightly larger region in [4]. It was shown in [4] that such a Banach algebra does not exist for . The method used in [1; 3; 4] was to prove the non-negativity of the expansion coefficients from which the existence of the Banach algebra easily follows. However, as shown in [4], the coefficients for a subset of can be negative infinitely often and so a different method must be used for these values of α and β. 
Askey, Richard; Gasper, George. Linearization of the Product of Jacobi Polynomials. III. Canadian journal of mathematics, Tome 23 (1971) no. 2, pp. 332-338. doi: 10.4153/CJM-1971-033-6
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[1] 1. Askey, R., Linearization of the product of orthogonal polynomials, pp. 223-228 in Problems in analysis (Princeton Univ. Press, Princeton, N.J., 1970). Google Scholar

[2] 2. Askey, R. and Wainger, S., A dual convolution structure for Jacobi polynomials, pp. 25–36 in Orthogonal expansions and their continuous analogues, Proc. Conference, Edwardsville, Illinois, 1967 (Southern Illinois Univ. Press, Carbondale, Illinois, 1968). Google Scholar

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

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[6] 6. Igari, S. and Uno, Y., Banach algebra related to the Jacobi polynomials, Töhoku Math. J. 21 (1969), 668–673. Google Scholar

[7] 7. Szegö, G., Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ., Vol. 23 (Amer. Math. Soc, Providence, R.I., 1967). Google Scholar

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