It is instructive to look at this from the stand-point of general solution, which is $ W(x) =\kappa \cdot \prod_{s=1}^n \frac{\Gamma\left(\frac{x-a_s}{h} + b_s + 1 \right)}{\Gamma\left(\frac{x-a_s}{h} +1 \right)} $
Using the recurrence equation for $\Gamma(z)$, i.e. $G(z+n) = (z+n-1) \cdots (z+1)z \Gamma(z)$, we get that $W(x)$ is polynomial iff if $b_s$ are non-negative integers.
added:
The necessary condition is that there exists a permutation of the tuple $\left[ b_1 +1 - \frac{a_1}{h}, \ldots, b_n +1 - \frac{a_n}{h}\right]$ so that it equals to $\left[m_1 + 1 - \frac{a_1}{h}, \ldots, m_n + 1 - \frac{a_n}{h}\right]$ for non-negative integers $m_1,\ldots,m_n$.
A non-trivial example was provided by @DidierPiau, with $n=2$, $h=1$, $[a_1,a_2] = [1,1/2]$, $[b_1,b_2] = [1/2,1/2]$. Then $[b_1+1-a_1/h,b_2+1-a_2/h] = [1/2,1]$, and $[1-a_1/h,1-a_2/h] = [0,1/2]$. The ratio of $\Gamma$-function, thus simplifies to $x$, and $W(x) = \kappa x$.