Closed-Form Solution Desired for Infinite Series

[stevendaryl]

Is there a simple closed-form solution for the following infinite series?

$F(a,b,c) = \sum_{j=0}^\infty \frac{(j+a)!}{(j+b)! (j+c)!}$

where $a, b, c$ are positive integers?

 

[haruspex]

Is there a simple closed-form solution for the following infinite series?

$F(a,b,c) = \sum_{j=0}^\infty \frac{(j+a)!}{(j+b)! (j+c)!}$

where $a, b, c$ are positive integers?

In the simple special case of a=b, that will give $e- \sum_{j=0}^{c-1} \frac 1 {j!}$, no?
I would not count that as closed form.