Consider the quantity $$ P(n,k) = \frac{ \Gamma ( n - k ) }{ \Gamma (n) \Gamma ( 1 - k ) } \left[ {}_3 F_2 \left( - \frac{1}{2} , 1 - n , - n ; 1 - k , \frac{3}{2} - n , - 1 \right) - 1 \right] ~, \qquad k , n \in {\mathbb Z}~. $$ for $1 \leq k \leq n-1$.
Written in the form above, it isn't obvious that this quantity is non-zero for $k \geq 1$ due to the $\Gamma(1-k)$ in the denominator, but some checks on Mathematica shows that there is also a $\Gamma(1-k)$ in the numerator that cancels out. There's also a $\Gamma(n-k)$ that appears in the denominator that cancels out that pre-factor. Further, $P(n,k)$ turns out to be a polynomial in $k$ of order $n-2$.
Given these simplifications, I would guess that there is a simpler form of the above polynomial where the factors $\Gamma(n-k)$ and $\Gamma(1-k)$ do not appear.
Can anyone guide me in the right direction towards finding such an expression?
My work - I am busy on Wolfram functions scanning through the large number of properties that hypergeometric functions satisfy to figure this out, but it's hard to figure out which property is useful here.