Asymptotic analysis of a family of polynomials associated with the inverse error function

TL;DR

This paper performs an asymptotic analysis of a polynomial family linked to the inverse error function, revealing their behavior for large degrees using advanced mathematical techniques.

Contribution

It introduces a detailed asymptotic analysis of polynomials related to the inverse error function, employing singularity analysis and WKB methods.

Findings

Derived asymptotic formulas for the polynomials as n→∞

Validated formulas with numerical results showing high accuracy

Connected polynomial behavior to derivatives of the inverse error function

Abstract

We analyze the sequence of polynomials defined by the differential-difference equation $P_{n+1}(x)=P_{n}^{\prime}(x)+x(n+1)P_{n}(x)$ asymptotically as $n\to\infty$. The polynomials $P_{n}(x)$ arise in the computation of higher derivatives of the inverse error function $\operatorname{inverf}(x)$. We use singularity analysis and discrete versions of the WKB and ray methods and give numerical results showing the accuracy of our formulas.