Asymptotics of Pseudo-Jacobi Polynomials with Varying Parameters

In this paper, we study the asymptotic behavior of the Pseudo-Jacobi polynomials P_n(z; a, b) as n →∞ 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 onn; more precisely, we assume a =−(An + A₀), A > 1andb = Bn + B₀, where A, B, A₀, B₀ are real constants. Our main tool is the asymptotic method developed for differential equations with a large parameter.