A note on the interlacing of zeros and orthogonality

Let {t(n)}(n=0)(infinity) be a sequence of monic polynomials with deg(t(n)) = n such that, for each n is an element of N, the zeros of t(n) are real and simple and t(n) and t(n+1) have no common zeros. We discuss the connection between the orthogonality of the sequence, the positivity of a certain ratio, and the interlacing of the zeros of t(n) and t(n+1) for n >= 1, n is an element of N. (C) 2008 Elsevier Inc. All rights reserved.