A necessary and sufficient condition for nonnegative product linearization of orthogonal polynomials
Denote by \(\{p_n\}_{n=0}^{N-1}\) a system of orthogonal polynomials with respect to a probability measure \(\mu\) on the real line. If the support of \(\mu\) is infinite \(N=+\infty\), otherwise \(N\) is the number of points in the support of \(\mu\). \noindent In this paper, the author investigates which \(\{p_n\}\) have the property \[ p_n(x)p_m(x)=\sum c(n,m,k)p_k(x) \] with nonnegative coefficients \(c(n,m,k)\) for every choice of \(n,m\) and \(k\). He shows that the nonnegative product linearization is equivalent to the existence of a family of special matrices with triangular support. The method goes through a discrete hyperpolic boundary value problem associated with the three-term recurrence relation. Such new result provides a unified approach to nonnegative linearization problems for orthogonal polynomial systems. Some examples are given for illustration.
scientific article; zbMATH DE number 2051471