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 P n ( 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 .