Asymptotic analysis of the Bell polynomials by the ray method

TL;DR

This paper develops asymptotic formulas for Bell polynomials as their degree grows large, using a discrete ray method applied to their differential-difference equation, with demonstrated accuracy through examples.

Contribution

It introduces a novel application of the discrete ray method to derive asymptotic approximations for Bell polynomials.

Findings

Asymptotic formulas closely match numerical values for large n

The differential-difference equation effectively models Bell polynomials

The method improves understanding of Bell polynomial behavior at large scales

Abstract

We analyze the Bell polynomials $B_{n}(x)$ asymptotically as $n\to\infty$. We obtain asymptotic approximations from the differential-difference equation which they satisfy, using a discrete version of the ray method. We give some examples showing the accuracy of our formulas.