EXPLICIT ORTHOGONAL POLYNOMIALS FOR RECIPROCAL POLYNOMIAL WEIGHTS ON ( 1; 1)

Let S be a polynomial of degree 2n + 2, that is positive on the real axis, and let w = 1=S on ( 1; 1). We present an explicit formula for the nth orthogonal polynomial and related quantities for the weight w. This is an analogue for the real line of the classical Bernstein-Szego formula for ( 1; 1). Orthogonal Polynomials, Bernstein-Szego formulas. 42C05 1. The Result 1 The Bernstein-Szego formula provides an explicit formula for orthogonal polynomials for a weight of the form p 1 x 2 =S (x) ; x 2 ( 1; 1) ; where S is a polynomial positive in ( 1; 1), possibly with at most simple zeros at 1. It plays a key role in asymptotic analysis of orthogonal polynomials. In this paper, we present an explicit formula for the nth degree orthogonal polynomial for weights w on the whole real line of the form (1.1) w = 1=S; where S is a polynomial of degree 2n + 2, positive on R. In addition, we give representations for the (n + 1)st reproducing kernel and Christo¤el function. We present elementary proofs, although they follow partly from the theory of de Branges spaces [1]. The formulae do not seem to be recorded in de Branges’book, nor in the orthogonal polynomial literature [2], [3], [7], [8], [9]. We believe they will be useful in analyzing orthogonal polynomials for weights on R. Recall that we may de…ne orthonormal polynomials fpmg n m=0, where (1.2) pm (x) = mx m +:::, m> 0; satisfying Z 1 pjpkw = 1 Because the denominator S in w has degree 2n + 2, orthogonal polynomials of degree higher than n are not de…ned. The (n + 1) st reproducing kernel for w is nX (1.3) Kn+1 (x; y) = pj (x) pj (y): j=0 Inasmuch as S is a positive polynomial, we can write jk