Large Degree Asymptotics of Generalized Bessel Polynomials

TL;DR

This paper derives new asymptotic expansions for generalized Bessel polynomials for large degrees, including simple, sector-based, and Bessel function forms, expanding on previous work.

Contribution

It introduces novel asymptotic expansions for generalized Bessel polynomials valid in different regions, including a simple expansion outside a neighborhood of the origin.

Findings

New simple expansion valid outside a compact neighborhood of the origin.

Asymptotic forms in sectors not containing turning points.

Expansion in terms of modified Bessel functions.

Abstract

Asymptotic expansions are given for large values of $n$ of the generalized Bessel polynomials $Y_n^\mu(z)$. The analysis is based on integrals that follow from the generating functions of the polynomials. A new simple expansion is given that is valid outside a compact neighborhood of the origin in the $z-$plane. New forms of expansions in terms of elementary functions valid in sectors not containing the turning points $z=\pm i/n$ are derived, and a new expansion in terms of modified Bessel functions is given. Earlier asymptotic expansions of the generalized Bessel polynomials by Wong and Zhang (1997) and Dunster (2001) are discussed.