restart;
with(SumTools):
an := -n+L;
bn := -x+L;
gen := Summation(Product(an, L=0..k-1)*Product(bn, L=0..k-1)/k!*(-1/a)^k, k=0..infinity);
genfun1 := subs(a=z,simplify(gen));
genfun2 := exp(z)*(1-(z/mu))^x;

correct is genfun2

but using famous hypergeom of charlier written in many books and papers, it can not summation to a correct generating function

i got two density, the part of it is similar the pattern of gamma, how to convert it into an expression of gamma times something?

test2 := x*Diff(P,x$2)+(a+1-x)*Diff(P,x)+mu*P=0;
Density := int(exp(-X*(I*u-a*i))*(-i*X)^(-a), X = -infinity .. infinity);

test2 := x*Diff(P,x$2)+(a+b-x)*Diff(P,x)+mu*P=0;
Density := int(-exp(-X*(I*u-a*i-b*i+i))*(-i*X)^(-a-b)*i*X, X = -infinity .. infinity);

use the formula provided in

http://en.wikipedia.org/wiki/Hermite_polynomials

and run in Maple

Restart;
with(SumTools):
Hn := n!*Summation(((-1)^(n/2-l))/((2*l)!*(n/2-l)!)*((2*x)^(2*l)),l=0..n/2);
genfun := Summation(Hn/n!*z^n,n=0..infinity);

but result is not exp(2*x*t - t^2)

1. I use your previous reply on V_G derive  on other characteristic function, but most are undefined

or complicated solution or can not evaluate, why?

for example

charc := 1+i*X/(i*X-1);
int(exp(-I*X*u)*charc, X = -infinity .. infinity);

it got this complicated thing
piecewise(Im(1/i) = 0, undefined, int(exp(-I*X*u)*(1+i*X/(i*X-1)), X = -infinity .. infinity, method = _UNEVAL))

Density := int(exp(-I*X*u)*charc, X = -infinity .. infinity);