Orthogonal polynomials and Padé approximants for reciprocal polynomial weights

Let ΓΓ be a closed oriented contour on the Riemann sphere. Let EE and FF be polynomials of degree n+1n+1, with zeros respectively on the positive and negative sides of ΓΓ. We compute the [n/n][n/n] and [n−1/n][n−1/n] Padé denominators at ∞∞ to f(z)=∫Γ1z−tdtE(t)F(t). As a consequence, we compute the nnth orthogonal polynomial for the weight 1/(EF)1/(EF). In particular, when ΓΓ is the unit circle, this leads to an explicit formula for the Hermitian orthogonal polynomial of degree nn for the weight 1/|F|21/|F|2. This complements the classical Bernstein–Szegő formula for the orthogonal polynomials of degree ≥n+1≥n+1.