Explicit Orthogonal Polynomials for Reciprocal Polynomial Weights

Let S be a polynomial of degree 2n + 2, that is, positive on the real axis, and let w = 1/S on (−∞,∞). 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-Szegő formula for (−1, 1). 1. The result The Bernstein-Szegő formula provides an explicit formula for orthogonal polynomials for a weight of the form √ 1− x2/S (x), x ∈ (−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 Christoffel 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 define orthonormal polynomials {pm}nm=0, where (1.2) pm (x) = γmx + · · · , γm > 0, satisfying ∫ ∞ −∞ pjpkw = δjk. Because the denominator S in w has degree 2n + 2, orthogonal polynomials of degree higher than n are not defined. The (n+ 1)st reproducing kernel for w is (1.3) Kn+1 (x, y) = n ∑