Linearization and connection coefficients of orthogonal polynomials

Abstract

Let {P n } =0/∞ be a system of orthogonal polynomials.Lasser [5] observed that if the linearization coefficients of {P n } =0/∞ are nonnegative then each of theP n (x) is a linear combination of the Tchebyshev polynomials with nonnegative coefficients. The aim of this paper is to give a partial converse to this statement. We also consider the problem of determining when the polynomialsP n can be expressed in terms ofQ n with nonnegative coefficients, where {Q n } =0/∞ is another system of orthogonal polynomials. New proofs of well known theorems are given as well as new results and examples are presented.