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.
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. 


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. 

