Can we get a closed form for the series


$\sum^\infty\_{k=0} \frac{ t^k}{k!} \Gamma(k+a)\Gamma(k+\frac{1}{2}){}\_2F\_1(k+a,k+\frac{1}{2};n+1,x)$


any hints or clues are welcomed.



infinite series with Hypergeometric functions
I too wonder about convergence. You can rewrite it as 
$$\Gamma \left( a\right) \Gamma \left( 1/2\right) \sum\_{k=0}^{\infty }\sum\_{j=0}^{\infty } \frac{\left( a\right) \_{j+k} \left( 1/2\right) \_{j+k}}{\left( n+1\right) \_{j}}\frac{t^{k}}{k!}\frac{x^{j}}{j!};$$
if you had an additional Pochhammer term indexed by k in the denominator, it would be Appel's $F\_{4}$ function.


