New Integral Identities for Orthogonal Polynomials on the Real Line

Let be a positive measure on the real line, with associated orthogonal polynomials fpng and leading coe¢ cients f ng. Let h 2 L1 (R) . We prove that for n 1 and all polynomials P of degree 2n 2, Z 1 1 P (t) pn (t) h pn 1 pn (t) dt = n 1 n Z 1 1 h (t) dt Z P (t) d (t) : As a consequence, we establish weak convergence of the measures in the lefthand side. Orthogonal Polynomials on the real line, Geronimus type formula, Poisson integrals 42C05 1. Introduction Let be a positive measure on the real line with in…nitely many points in its support, and R xd (x) …nite for j = 0; 1; 2; ::: . Then we may de…ne orthonormal polynomials pn (x) = nx n + :::, n > 0; satisfying Z 1 1 pnpmd = mn: Let (1.1) Ln (x; y) = n 1 n (pn (x) pn 1 (y) pn 1 (x) pn (y)) and for non-real a; (1.2) En;a (z) = s 2 jLn (a; a)j Ln ( a; z) : In a recent paper [6], we used the theory of de Branges spaces [1] to show that for Im a > 0, and all polynomials P of degree 2n 2, we have (1.3) Z 1 1 P (t) jEn;a (t)j dt = Z P (t) d (t) : This may be regarded as an analogue of Geronimus’ formula for the unit circle, where instead of En;a, we have a multiple of the orthonormal polynomial on the Date : May 18, 2010. 1Research supported by NSF grant DMS0700427 and US-Israel BSF grant 2008399 1