Consider the sum $$Q(v,u) = \sum_{l=0}^\infty\sum_{k=0}^{l}\frac{u^l}{l!} \frac{v^k}{k!}$$ which arises from the inverse Laplace transform of $f(s) = \frac{1}{s(s-a)}e^{b/s}.$
Is there a means to express $Q(v,u)$ in terms of some special functions? It seems to be some sort of incomplete Humbert series like $$ \Phi_3(\beta,\gamma,x,t) = \sum_{m=0}^\infty \sum_{n=0}^\infty \frac{(\beta)_m}{(\gamma)_{m+n}m!n!}x^my^n$$ in that it has the wrong summation limits. Any thoughts are greatly appreciated!
On
$\sum\limits_{l=0}^\infty\sum\limits_{k=0}^l\dfrac{u^lv^k}{l!k!}$
$=\sum\limits_{k=0}^\infty\sum\limits_{l=k}^\infty\dfrac{u^lv^k}{l!k!}$
$=\sum\limits_{k=0}^\infty\sum\limits_{l=0}^\infty\dfrac{u^{l+k}v^k}{(l+k)!k!}$
$=\Phi_3(1,1;u,uv)$ (according to https://en.wikipedia.org/wiki/Humbert_series)
As I showed in my answer to your other question Closed form for an infinite series involving lower incomplete gamma functions,
$Q(u, v)+Q(v, u) =e^{u+v}+I_0(2\sqrt{uv}) $.
I also suggested that you research the Marcum Q-function.