Riemann-Hilbert analysis for Jacobi polynomials orthogonal on a single contour

Classical Jacobi polynomials P_n^(α,β), with α, β > - 1, have a number of well-known properties, in particular the location of their zeros in the open interval (-1, 1). This property is no longer valid for other values of the parameters; in general, zeros are complex. In this paper we study the strong asymptotics of Jacobi polynomials where the real parameters α_n, β_n depend on n in such a way that lim n→∞ α_n/n = A, lim n→∞ β_n/n = B, with A, B ∈ ℝ. We restrict our attention to the case where the limits A, B are not both positive and take values outside of the triangle bounded by the straight lines A = 0, B = 0 and A + B + 2 = 0. As a corollary, we show that in the limit the zeros distribute along certain curves that constitute trajectories of a quadratic differential. The non-hermitian orthogonality relations for Jacobi polynomials with varying parameters lie in the core of our approach; in the cases we consider, these relations hold on a single contour of the complex plane. The asymptotic analysis is performed using the Deift-Zhou steepest descent method based on the Riemann-Hilbert reformulation of Jacobi polynomials.