There exist many integral representations of the generalised hypergeometric function ${}_3 F_2$ but these are all assuming that the entries are positive integers. I have a problem involving,
$${}_3F_2 \left(-n, 1+\frac{a}{2}, -\frac{a}{2}; 1, -b; 1\right)$$
where $0 \leq n \leq \lfloor \frac{b}{2} \rfloor$, $b \geq \frac{a}{2}$ and $a$ is even. In this case it is well-defined, since the numerator reaches zero before the denominator, and you get a polynomial in $n$. However, I can't use any of these integral representations since things like $\Gamma(-b)$ appear. Are there integral representations that allow for negative integer entries in the hypergeometric function?
As an example of what goes wrong, if you look at the Euler's integral transform it would give,
$${}_3F_2 \left(-n, 1+\frac{a}{2}, -\frac{a}{2}; 1, -b; 1\right) = \frac{\Gamma(-b)}{\Gamma(-a/2)\Gamma(a/2-b)} \int_0^1 {}_2 F_1(\dots) \dots$$
and that prefactor evaluates to zero for even $a$ (or indeterminate if you evaluate $b$ first) which is obviously not right since ${}_3 F_2$ is not zero.