Geodesic
The Linearization of the Product of Continuous q-Jacobi Polynomials
Canadian journal of mathematics, Tome 33 (1981) no. 4, pp. 961-987

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

The problem of linearizing the product of two Jacobi polynomials, Pm (α, β)(x)Pn (α, β)(x), and to establish the conditions for the non-negativity of the coefficients has been of considerable interest for many years. Explicit non-negative representations were sought and found by many authors [7, 8, 13, 14], but only in the special case α = β, although Hylleraas [14] succeeded in finding a formula in another case α = β + 1. Gasper [9, 10] found the necessary and sufficient conditions for the non-negativity of the linearization coefficients by exploiting a recurrence relation obtained by Hylleraas for the above-mentioned product. Koornwinder [16] approached the same problem from a different point of view and managed to find a non-negative integral expression to these coefficients when . However, an exact formula in a hypergeometric series form for general α, β has been very elusive so far, in spite of the fact that all computation of special cases seemed to indicate that such a formula should exist. 
Rahman, Mizan. The Linearization of the Product of Continuous q-Jacobi Polynomials. Canadian journal of mathematics, Tome 33 (1981) no. 4, pp. 961-987. doi: 10.4153/CJM-1981-076-8
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