I am trying to find the closed form for the following
\begin{equation} {_2F_2}(2,4;3,4+a;a) \end{equation}
any help would be greatly appreciated. thank you
I find the closed form for \begin{equation} {_2F_2}(1,3;2,3+a;a)=1+\frac{a}{2} \end{equation} But don't success to solve the first sum
$_2F_2(2,4;3,4+a;a)$
$=\sum\limits_{n=0}^\infty\dfrac{(2)_n(4)_na^n}{(3)_n(a+4)_nn!}$
$=\sum\limits_{n=0}^\infty\dfrac{(n+3)(2)_na^n}{3(a+4)_nn!}$
$=\sum\limits_{n=0}^\infty\dfrac{(n+2)(2)_na^n}{3(a+4)_nn!}+\sum\limits_{n=0}^\infty\dfrac{(2)_na^n}{3(a+4)_nn!}$
$=\sum\limits_{n=0}^\infty\dfrac{(2)_{n+1}a^n}{3(a+4)_nn!}+\sum\limits_{n=0}^\infty\dfrac{(2)_na^n}{3(a+4)_nn!}$
$=\sum\limits_{n=0}^\infty\dfrac{2(3)_{n}a^n}{3(a+4)_nn!}+\sum\limits_{n=0}^\infty\dfrac{(2)_na^n}{3(a+4)_nn!}$
$=\dfrac{2}{3}{}_1F_1(3;a+4;a)+\dfrac{1}{3}{}_1F_1(2;a+4;a)$
$=\dfrac{(a+3)(a+2)(a+1)}{3}\int_0^1x^2(1-x)^ae^{ax}~dx+\dfrac{(a+3)(a+2)}{3}\int_0^1x(1-x)^{a+1}e^{ax}~dx$
Which is expected to related to incomplete gamma function