Uniform Asymptotic approximation and numerical evaluation of the Reverse Generalized Bessel Polynomial zeros

Abstract

Uniform asymptotic expansions are derived for the zeros of the reverse generalized Bessel polynomials of large degree $n$ and real parameter $a$. It is assumed that $-\Delta_{1} n+\frac{3}{2} \leq a \leq \Delta_{2} n$ for fixed arbitrary $\Delta_{1} \in (0,1)$ and bounded positive $\Delta_{2}$. For this parameter range at most one of the zeros is real, with the rest being complex conjugates. The new expansions are uniformly valid for all the zeros, and are shown to be highly accurate for moderate or large values of $n$. They are consequently used as initial values in a very efficient numerical algorithm designed to obtain the remaining complex zeros using Taylor series.