Wiman-Valiron theory for a polynomial series based on the Wilson operator

TL;DR

This paper develops a Wiman-Valiron theory for entire functions expanded in Wilson series, providing local polynomial-like behavior estimates and growth analysis of solutions to Wilson difference equations.

Contribution

It introduces a novel Wiman-Valiron framework for Wilson series, extending classical theory to this operator and analyzing entire solutions of related difference equations.

Findings

Functions of order less than 1/3 behave like local polynomials near their maximum term.

Estimates of Wilson operator derivatives compared to the functions themselves.

Application to growth analysis of solutions to Wilson difference equations.

Abstract

We establish a Wiman-Valiron theory for a polynomial series based on the Wilson operator $\mathcal{D}_\mathrm{W}$. For an entire function $f$ of order smaller than $\frac13$, this theory includes (i) an estimate which shows that $f$ behaves locally like a polynomial consisting of the terms near the maximal term in its Wilson series expansion, and (ii) an estimate of $\mathcal{D}_\mathrm{W}^n f$ compared to $f$. We then apply this theory in studying the growth of entire solutions to difference equations involving the Wilson operator.