Convolution polynomials

TL;DR

This paper explores polynomials derived from power series raised to a variable power, revealing their unique properties and providing methods for recognition, application, and asymptotic approximation.

Contribution

It introduces a framework for understanding and utilizing convolution polynomials, including a new general asymptotic approximation result.

Findings

Identification of key properties of convolution polynomials

Methods for recognizing these polynomials in various contexts

A new asymptotic approximation theorem for convolution polynomials

Abstract

The polynomials that arise as coefficients when a power series is raised to the power $x$ include many important special cases, which have surprising properties that are not widely known. This paper explains how to recognize and use such properties, and it closes with a general result about approximating such polynomials asymptotically.