Reduction of hypergeometric function for integer parameters Consider integers $a,b,c,d>0$ and the hypergeometric function
$${}_3F_2(1-d,b,a+b;a+b+c,b+1;1)$$
I don't know much about hypergeometric functions but I understand that when the parameters are integer valued we get some simplification. Unfortunately, I can't seem to figure this out or find a good reference.

My attempt:
For $d=1$, the function evaluates to 1.
For $d=2$, we have
$$1-\frac{b(a+b)}{(b+1)(a+b+c)}$$
For $d=3$, we have
$$1-\frac{2b(a+b)}{(b+1)(a+b+c)} + \frac{b(a+b)(a+b+1)}{(b+2)(a+b+c)(a+b+c+1)}$$
For $d=4$, we have
$$1-\frac{3b(a+b)}{(b+1)(a+b+c)} + \frac{3b(a+b)(a+b+1)}{(b+2)(a+b+c)(a+b+c+1)} - \frac{b(a+b)(a+b+1)(a+b+2)}{(b+3)(a+b+c)(a+b+c+1)(a+b+c+2)}$$
I can begin to write out the expression for general integer $d\ge 1$ as
$$\sum_{k=0}^{d-1}(-1)^k\binom{d-1}{k}\cdots$$
I'm not sure how the rest proceeds.

Wolfram functions reference is a good place to find identities like this. Here's the one for $\;_3F_2$. For this, you probably want. Unfortunately I'm not finding much in values at $z = 1$, but there's a lot to go through.
