With Maple help I solve the problem:
HoldForm[Sum[(E^-l l^n n (1 - x)^(-m + n) x^m)/(
m m! (-m + n)!), {n, 0, Infinity}, {m, 1,
Infinity}] == -Exp[-l*x]*(-1 + l*(-1 + x))*l*x*
HypergeometricPFQ[{1, 1, -l*x + l + 2}, {2, 2, -l*x + l + 1},
l*x]] // TeXForm
$$\sum _{n=0}^{\infty } \sum _{m=1}^{\infty } \frac{e^{-l} l^n n (1-x)^{-m+n} x^m}{m m! (-m+n)!}=-\exp (-l x)
(-1+l (-1+x)) l x \, _3F_3(1,1,-l x+l+2;2,2,-l x+l+1;l x)$$
For:
$\{-1<x<1,l\in \mathbb{R}\}$
ff[x_, l_, M_] :=
Sum[(E^-l l^n n (1 - x)^(-m + n) x^m)/(
m m! (-m + n)!), {n, 0, M}, {m, 1, M}] // N
f[x_, l_, M_] := Sum[-((E^-l l^n n (1 - x)^n x HypergeometricPFQ[{1, 1, 1 - n}, {2, 2},
x/(-1 + x)])/((-1 + x) (-1 + n)!)), {n, 0, M}] // N;
g[x_, l_] := -Exp[-l*x]*(-1 + l*(-1 + x))*l*x*HypergeometricPFQ[{1, 1, -l*x + l + 2}, {2, 2, -l*x + l + 1}, l*x]
ff[1/2, -1, 20]
(*-0.282835*)
f[1/2, -1, 20]
(*-0.282835*)
g[1/2, -1] // N
(*-0.282835*)
