Problem 4, Chapter 9 in "Special Functions and Their Applications" (N.N. Lebedev)is to show that
$$w(z,t)=(1-t)^{\beta -\gamma } (1-t+z t)^{-\beta }=\sum _{n=0}^{\infty } \frac{(\gamma) _n \, _2F_1(-n,\beta ;\gamma ;z) t^n}{n!}$$ $| t| <\min \left\{1,\frac{1}{| z-1| }\right\}$
Following the usual argument, we can expand $w (z, t )$ on a Taylor series about $ t =0$. $w(z,t)=\sum _{n=0}^{\infty } c_n(z) t^n$ where $$c_n(z)=\int \frac{w(z,t)}{2 \pi i t^{n+1}} \, dt$$ where the integration is along any circle of sufficiently small radius enclosing the point $t =0$. Inserting the expression for $w (z,t)$, we see that the integrand is very similar to that appearing in the well known integral formula for the Hypergeometric function. This later formula however is a definite integral over the real axis from 0 to 1.
Employing the theory of residues we have $$n! c_n(z)=\underset{t\to 0}{\text{lim}}\frac{\partial ^n\left((1-t)^{\beta -\gamma } (1-t (1-z))^{-\beta }\right)}{\partial t^n}$$
In the case of the classic orthogonal polynomials (Legendre, Laguerre etc.), one would appeal to the appropriate Rodrigues formula at this stage. I am unaware of such type formula for the Hypergeometric function. Proceeding to evaluate the nth derivative by Leibnitz's rule, we obtain
$$n! c_n(z)=(-1)^n \sum _{k=0}^n \binom{n}{k} (-\beta )^{(k)} (\beta -\gamma )^{(n-k)} (1-z)^k$$ where $x^{(k)}=(x (x-1) (x-2)\text{...}) (x-(k-1))$
Hence, we must show that $$(\gamma )_n \, _2F_1(-n,\beta ;\gamma ;z)=(-1)^n \sum _{k=0}^n \binom{n}{k} (-\beta )^{(k)} (\beta -\gamma )^{(n-k)} (1-z)^k$$
It can be shown that $$\, _2F_1(-n,\beta ;\gamma ;z)=\sum _{k=0}^n \frac{(-1)^k \binom{n}{k} (\beta )_k z^k}{(\gamma )_k}$$
How do we show that these two polynomials are identical ? I tried a proof by induntion, but was not successful.