Linearization and connection coefficients of polynomial sequences: A matrix approach

For any sequence of polynomials {p(k) (t)} in one real or complex variable, where p(k) has degree k for k >= 0, we find explicit expressions and recurrence relations for infinite matrices whose entries are the numbers d(n, m, k), called linearization coefficients, that satisfy p(n) (t)p(m) (t) = Sigma(n+m)(k=0) d(n, m, k)p(k) (t), n, m >= 0. For any pair of polynomial sequences {u(k) (t)} and {p(k) (t)} we find infinite matrices whose entries are the numbers e(n, m, k) that satisfy p(n) (t)p(m) (t) = Sigma(n+m)(k=0) e(n, m, k)u(k) (t), n, m >= 0. We also obtain recurrence relations and other properties of the linearization coefficients. Such results are obtained using only simple algebraic properties of infinite matrices. We apply the general results to general orthogonal polynomial sequences and to some simple families of orthogonal polynomials, such as the Chebyshev, Hermite, and Charlier families. (c) 2023 Elsevier Inc. All rights reserved.