Let $n,N,T$ be positive integers, with $N=\binom{n}{2}$, and $3\leq n\leq T\leq N$. Define:

$$P(z):=z^{N+1-T}\int_0^1\frac{(1-t^2)^{n-2}}{(1-(1-z)t)^{N+1}}{}_2F_1\left[-T,N+1,N+2-T; \frac{tz}{1-(1-z)t}\right]dt.$$

I am interested in evaluating the limit $\lim_{z\rightarrow 0}P(z)$. I am wondering if there is any nice expression for this limit, perhaps as another hypergeometric function. I would be thrilled if the limit is another integral. In fact, I know that $P(z)$ itself is a *polynomial*, so there's a lot of magic happening to make this work.

The difficulty is resolving the singularity at $z=0,t=1$. It seems like all of the terms in the hypergeometric function contribute to this integral. Notice that one cannot just write the hypergeometric function as a sum and then swap the swap the sum with the integral *and* the limit. I've tried a number of Kummer transformations to somehow make things nicer but I haven't see much progress. I've consulted a number of integral tables but haven't seen such an integral come up.

One thing I've noticed is that since $-T$ is a negative integer, the hypergeometric function is a finite polynomial. In fact, it is proportional to the Jacobi polynomial $P_T^{(N+1-T,-1)}\left[\frac{1-t(1+z)}{1-t(1-z)}\right]$. I mention this because the $(1-t^2)^{n-2}$ term reminds me of the orthogonality weight for certain Jacobi polynomials, so perhaps some nice inner product is being computed here? Moreover, the full ratio $(1-t^2)^{n-2}/(1-(1-z)t)^{N+1}$ looks like the primary term involved in the Schlafli integral representation for Associated Legendre polynomials, although there the integral is on $[-1,1]$ or on a contour surrounding $1/(1-z)$.

I was thinking of splitting the integral into two pieces: $\int_0^1=\int_0^c+\int_c^1$, with hopes that if $c$ is chosen carefully (as a function of $z$), only the latter integral contributes. It baffles me though, how to obtain some kind of uniform bound on the hypergeometric part with $z\approx 0$ and $t\approx 1$. Is there perhaps some modification of steepest descent or saddle point methods that would apply here?