Uniform asymptotics for Jacobi polynomials with varying large negative parameters-- a Riemann-Hilbert approach

An asymptotic expansion is derived for the Jacobi polynomials $P_{n}^{(\alpha_{n},\beta_{n})}(z)$ with varying parameters $\alpha_{n}=-nA+a$ and $\beta_n=-nB+b$, where $A>1, B>1$ and $a,b$ are constants. Our expansion is uniformly valid in the upper half-plane $\overline{\mathbb{C}}^+=\{z:{Im}\; z \geq 0\}$. A corresponding expansion is also given for the lower half-plane $\overline{\mathbb{C}}^-=\{z:{Im}\; z \leq 0\}$. Our approach is based on the steepest-descent method for Riemann-Hilbert problems introduced by Deift and Zhou (1993). The two asymptotic expansions hold, in particular, in regions containing the curve $L$, which is the support of the equilibrium measure associated with these polynomials. Furthermore, it is shown that the zeros of these polynomials all lie on one side of $L$, and tend to $L$ as $n \to \infty$.