Common zeros of two polynomials in an orthogonal sequence

We show that for any positive integers k < m there exists a sequence p(0), .... p(m) of orthogonal polynomials (p(i) having degree i) such that p(k) a nd p(m) have min{k, m-k-l} zeros in common, the maximum possible. More gene rally, if, in a sequence p(0), ..., p(m) of orthogonal polynomials, p(k) an d p(m) have no common zero, then for every n (m + 1 less than or equal to n less than or equal to m + k), there exists an orthogonal sequence q(0), .. ., q(n) such that (i) q(k) = p(k) and (ii) the zeros of q(n) are precisely the zeros of p(m) together with n-m zeros of P-k. (C) 2000 Academic Press.