I've encountered the following sum: $$ c_m(x;a)=\sum_{n=0}^{m-1} \frac{\Gamma(a+n)}{\Gamma(a+m)} x^{m-n-1} = \sum_{n=0}^{m-1} B(a+n,m-n) \frac{x^{m-n-1}}{(m-n-1)!}, \qquad a,x>0. $$
It really seems like it is related to some special function, and indeed Mathematica will (after some cleaning up) generate a somewhat messy relation with respect to upper-incomplete gamma functions: $$ c_m(x;a) = (-x)^{m+a-1} \mathrm{e}^{-x} \left[ \Gamma(-(m+a-1),-x) - (-1)^m \frac{\Gamma(a)}{\Gamma(m+a)}\Gamma(1-a,-x)\right]. $$
Does anybody have any insight where this is coming from? Hypergeometric functions and/or Jacobi polynomials are not my specialty, but I can only imagine that it is rooted in those definitions somehow.
The result you obtain can also be wrtitten as $$c_m(x;a)=\sum_{n=0}^{m-1} \frac{\Gamma(a+n)}{\Gamma(a+m)} x^{m-n-1} = \frac{e^{-x} \left[\Gamma (a+m) E_{a+m}(-x)-\Gamma (a) x^m E_a(-x)\right]}{\Gamma (a+m)}$$ where appear the exponential integral function