Simplified Airy function Asymptotic expansions for Reverse Generalised Bessel Polynomials

TL;DR

This paper derives simplified uniform asymptotic expansions for reverse generalized Bessel polynomials of large degree, using Airy functions and Liouville-Green methods, applicable in complex domains and near turning points.

Contribution

It introduces more straightforward asymptotic expansions for these polynomials, involving explicit recursive coefficients and uniform validity across complex arguments and parameter ranges.

Findings

Derived simpler asymptotic expansions involving Airy functions.

Provided explicit recursive formulas for coefficient functions.

Achieved uniform validity in complex domains and near turning points.

Abstract

Uniform asymptotic expansions are derived for reverse generalised Bessel polynomials of large degree $n$, real parameter $a$, and complex argument $z$, which are simpler than previously known results. The defining differential equation is analysed; for large $n$ and $\frac{3}{2} - n < a < \infty$, it possesses two turning points in the complex $z$ plane which are complex conjugates. Away from these turning points Liouville-Green expansions are obtained for the polynomials and two companion solutions of the differential equation, where asymptotic series appear in the exponent. Then representations involving Airy functions and two slowly varying coefficient functions are constructed. Using the Liouville-Green representations, asymptotic expansions are obtained for the coefficient functions that involve coefficients that can be easily and explicitly computed recursively. In conjunction with a suitable re-expansion, or Cauchy's integral formula, near the turning point, the expansions are valid for $-\Delta_{1} n+\frac{3}{2} \leq a \leq \Delta_{2} n$ for fixed arbitrary $\Delta_{1} \in (0,1)$ and bounded positive $\Delta_{2}$, uniformly for all unbounded complex values of $z$.