I've been dealing with sums and integrals over hypergeometric functions quite a bit lately, and the latest problem is the following double sum: \begin{equation} F(x,y;\alpha,t)=\sum_{n,m=0}^\infty\frac{x^ny^m}{n!m!}\frac{\Gamma(\alpha+n+m+z-1)}{\Gamma(\alpha+n+m)}{}_2F_1(-n,-m;1/2;t), \end{equation} where $x\geq1$, $y\geq1$, $1\geq t\geq0$, $1<\alpha\leq2$, and $1-\alpha<\Re(z)<0$. One of the routes I've taken is trying to use the series definition (over all $k\geq0$) of the hypergeometric function and change the order of summation. This makes sense to me, since the sum is bounded by the values at $t=0$ and $t=1$, for each of which it is easy to sum over at least one of $n$ or $m$, finding two distinct results for $t=0,1$. Since the inner sum over $n$ or $m$ is convergent, switching the order of that sum and the $k$-sum from the ${}_2F_1$ should be allowed...?
However, the only way I know how to then sum over $n$ (wlog), is to re-express everything in terms of Pochhammer symbols with index $n$. When I do this for the $(-n)_k$ piece, I find \begin{equation} (-n)_k=(-1)^k(n-k+1)_k=(-1)^k\frac{(1)_n}{(1-k)_n}(1-k)_k=\frac{(1)_n}{(1-k)_n}(0)_k, \end{equation} which projects out only the $k=0$ term, which is independent of $t$. This contradicts the difference in results for $t=0,1$.
Where have I gone wrong here? I imagine it's in the second equivalence above, where I have implicitly multiplied by $1=\Gamma(1-k)/\Gamma(1-k)$, which is undefined for $k\neq0$, but tends to one in all limits. If this is indeed illegal, is there a trick to perform this sum? I see that it is very close to the only summation formula for ${}_2F_1$ on functions.wolfram.com, but the Pochhammer symbol is all wrong.