Orthogonal polynomials and Padé approximants for reciprocal polynomial weights

摘要

Let Γ 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 Γ. We compute the [n / n] and [n - 1 / n] Padé denominators at ∞ to f (z) = ∫_Γ frac(1, z - t) frac(d t, E (t) F (t)). As a consequence, we compute the nth orthogonal polynomial for the weight 1 / (E F). In particular, when Γ is the unit circle, this leads to an explicit formula for the Hermitian orthogonal polynomial of degree n for the weight 1 / | F |~2. This complements the classical Bernstein-Szego{double acute} formula for the orthogonal polynomials of degree ≥ n + 1.