  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.
