Do you mean you want to derive the generating function for the Laguerre polynomials from the differential equation? By the way, that web page is wrong: it should be an ordinary generating function
not an exponential generating function with n! in the denominator.
I don't know if that is possible. I can derive a partial differential equation for the generating function, but Maple doesn't seem able to find the desired solution.
The Laguerre polynomial Ln(x) is the solution of the differential equation x y" + (1-x) y' + n y = 0 with initial conditions y(0) = 1 (the differential equation has a regular singular point at x=0 and there is only a one-parameter family of solutions that have finite limits at 0).
Let the generating function be
The initial conditions say
> g(t,0) = sum(t^n*1, n=0..infinity);
Sum the first and second terms from n=0 to infinity, and you get
But for the third term, noting that
So g should satisfy the PDE
pdsolve can solve the PDE, but not with the boundary condition g(t,0) = 1/(1-t).
Here's a different approach. Maple has the LaguerreL function. It knows that this is a solution of the Laguerre differential equation:
> dsolve(x*diff(y(x),x$2) + (1-x)*diff(y(x),x) + n*y(x));
> convert(%, LaguerreL);
And it knows a power series expansion for this:
> convert(LaguerreL(n,x), FormalPowerSeries,x);
Actually the sum should be for k from 0 to n because pochhammer(-n,k) = 0 if n is an integer and k > n: after all, LaguerreL(n,x) is supposed to be a polynomial of degree n in x. So the generating function should be
Interchange the order of summations and you have
Maple needs a little help to do the sum. We need to use an identity relating pochhammer(-n,k) to pochhammer(k+1,n-k).
> convert(pochhammer(-n,k)/pochhammer(k+1,n-k), factorial) assuming n::posint, k::posint, n>=k;
So now our double sum becomes
> sum(sum( pochhammer(k+1,n-k)*(-1)^k/(k!*(n-k)!) * x^k * t^n, n=k..infinity), k=0..infinity);