Closed form for this power series looking like an hypergeometric?

Question

I would like to "resum" the following expression:

$$\sum_{k=0}^a \frac{(-a)_k (-b)_k}{(c)_k} x^k\,, \tag{1}$$

with $a, b, c$ positive even numbers and $x > 0$ real. Is there a known function for this finite sum? Wolfram Alpha and Mathematica return a DifferenceRoot system.

Note that if there was a $k!$ in the denominator, this would be a hypergeometric function $_2F_1(a,b,c,x)$, as explained here.

Answer 1

Okay I am stupid: This can be brought into the form of a generalised hypergeometric function by multiplying the argument of the sum by $\frac{k!}{k!} = \frac{(1)_k}{k!}$, and then we find that eq. $(1)$ is the same as:

$$_3F_1 (-a,-b,1;c;x)\,. \tag{2}$$