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

Classical Jacobi polynomials , 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 This property is no longer valid for other values of the parameters; in general, zeros are complex.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

with . 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.