Change of Basis for Products of Orthogonal Polynomials
Stephen Mark Barnett · SIAM Journal on Algebraic and Discrete Methods · 1987
Given two sets of orthogonal polynomials $\mathcal{B} = \{ p_i (\lambda )\} $ and $\{ q_i (\lambda )\} $, simple procedures are given for expressing the products $\lambda ^i p_j (\lambda ),\lambda ^i q_j (\lambda ),p_i (\lambda )q_j (\lambda )$ and $q_i (\lambda )q_j (\lambda )$ in terms of the basis $\mathcal{B}$. The main computations involved are multiplications of vectors by a tridiagonal matrix. The results are based on a previous theorem for determining the product of two polynomials both expressed relative to $\mathcal{B}$.