Recurrence equations involving different orthogonal polynomial sequences and applications

Abstract

Consider {p n } ∞ n=0 , a sequence of polynomials orthogonal with respect to w(x) > 0 on (a, b), and polynomials {g n,k } ∞ n=0 , k ∈ N 0 , orthogonal with respect to c k (x)w(x) > 0 on (a, b), where c k (x) is a polynomial of degree k in x. We show how Christoffel's formula can be used to obtain mixed three-term recurrence equations involving the polynomials p n , p n−1 and g n−m,k , m ∈ {2, 3, . . . , n−1}. In order for the zeros of p n and G m−1 g n−m,k to interlace (assuming p n and g n−m,k are co-prime), the coefficient of p n−1 , namely G m−1 , should be of exact degree m − 1, in which case restrictions on the parameter k are necessary. The zeros of G m−1 can be considered to be inner bounds for the extreme zeros of the (classical or q-classical) orthogonal polynomial p n and we give examples to illustrate the accuracy of these bounds. Because of the complexity the mixed three-term recurrence equations in each case, algorithmic tools, mainly Zeilberger's algorithm and its q-analogue, are used to obtain them.