The Linearization of the Product of Continuous q-Jacobi Polynomials

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Abstract
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.

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