Linearization of the Product of Jacobi Polynomials. III

Abstract

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.