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

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 $\frac{1}{3}$, 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.