Old and New Geronimus Type Identities for Real Orthogonal Polynomials

Let be a positive measure on the real line, with orthogonal polynomials fpng and leading coe¢ cients f ng. The Geronimus type identity 1 jIm zj Z 1 1 P (t) jzpn (t) pn 1 (t)j dt = n 1 n Z P (t) d (t) ; valid for all polynomials P of degree 2n 2 has known analogues within the theory of orthogonal rational functions, though apparently unknown in the theory of orthogonal polynomials. We present new proofs of this and its generalization, 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) ; valid for any h 2 L1 (R) : Orthogonal Polynomials on the real line, Geronimus type formula, Poisson integrals 42C05 1. Introduction In the theory of orthogonal polynomials on the unit circle, Geronimus’identity [5, p. 198] plays an important role. Recall that if is a …nite positive Borel measure on the unit circle, with in…nitely many points in its support, and orthonormal polynomials f ng, so that 1 2 Z 2 0 n e m (ei )d ( ) = mn; then Geronimus’identity asserts that Z 2 0 P e j n (ei )j d = Z P e d e ; for all polynomials P of degree n. By symmetry, this extends to P that is a Laurent polynomial. Geronimus’identity is very useful in asymptotics for orthogonal polynomials on the unit circle, see the books of Freud [5] and Simon [12]. As far as the author was aware, there was no known analogue for orthogonal polynomials on the real line. At least, none is mentioned in the classical textbooks on orthogonal polynomials. While using the theory of de Branges spaces in the context of universality limits for random matrices, the author discovered such an identity. Date : October 20, 2010. 1Research supported by NSF grant DMS0700427 and US-Israel BSF grant 2008399 1