Asymptotics of Pseudo-Jacobi Polynomials with Varying Parameters

In this paper, we study the asymptotic behavior of the Pseudo-Jacobi polynomials Pn(z; a, b) as n -> 8 for z in the whole complex plane. These polynomials are also known as the Romanovski-Routh polynomials. They occur in quantum mechanics, quark physics, and random matrix theory. When the parameter a is fixed or a > -n, there is no real-line orthogonality. Here, we consider the case when the parameters a and b depend on n; more precisely, we assume a = -(An + A(0)), A > 1 and b = Bn + B-0, where A, B, A(0), B-0 are real constants. Our main tool is the asymptotic method developed for differential equations with a large parameter.