Orthogonal polynomials and Padé approximants for reciprocal polynomial weights

Let be a closed oriented contour on the Riemann sphere. Let E and F be polynomials of degree n + 1, with zeros respectively on the positive and negative sides of . We compute the [n=n] and [n 1=n] Padé denominator at 1 to f (z) = Z 1 z t dt E (t)F (t) : As a consequence, we compute the nth orthogonal polynomial for the weight 1= (EF ). In particular, when is the unit circle, this leads to an explicit formula for the Hermitian orthogonal polynomial of degree n for the weight 1= jF j. This complements the classical Bernstein-Szeg1⁄2o formula for the orthogonal polynomials of degree n+ 1: Padé approximant, de Branges space, reproducing kernel, Orthogonal Polynomials, Bernstein-Szeg1⁄2o formula. AMS Classi…cation: 41A21, 42C99 Research supported by NSF grant DMS0400446 and US-Israel BSF grant 2004353 1. The Result Let f be a formal power series at 1 of the form