Partial sum of Hypergeometric2F1 function Can we get a closed form for the following series:
$$\sum\limits_{x=1}^c \dfrac{(x+c-1)!}{x!} {}_2F_1(x+c,x,x+1,z)$$
where $c$ is a positive integer and $z$ is a real number less than -1?
Any suggestions or hints are appreciated.

$$\sum\limits_{x=1}^c \dfrac{(x+c-1)!}{x!} {}_2F_1(x+c,x,x+1,z)=\frac{(c-1)!}{(z-1)^{2c-1}}P_c(z),$$
where $P_c(z)$ is a polynomial in $z$ of degree $2c-2$, the first few are
$$P_1(z)=1$$
$$P_2(z)=z^2-4 z+5$$
$$P_3(z)=z^4-6 z^3+16 z^2-24 z+19$$
$$P_4(z)=z^6-8 z^5+29 z^4-64 z^3+97 z^2-104 z+69$$
$$P_5(z)=z^8-10 z^7+46 z^6-130 z^5+256 z^4-380 z^3+446 z^2-410 z+251$$
$$P_6(z)=z^{10}-12 z^9+67 z^8-232 z^7+562 z^6-1024 z^5+1484 z^4-1792 z^3+1847 z^2-1572 z+923$$
$$P_7(z)=z^{12}-14 z^{11}+92 z^{10}-378 z^9+1093 z^8-2380 z^7+4096 z^6-5810 z^5+7071 z^4-7630 z^3+7344 z^2-5992 z+3431$$
these polynomials may well have appeared before...
