Question: How to solve this eigenfunction

i use Infeld's The Factorization method and find this Hermit is type D. since not exactly match r i use R(x,m) = r(x,m) + f(m) and compare them, then find k = x but eigenfunction Y := C*exp(-int(x,x)); has constant C, range x should between negative infinity and infinity. C is 1/Pi^(1/4) after solved. i guess this is wrong, but i do not know where is wrong.

another problem is that i assume C = 1, after sum for generating function the solution is not the same as mentioned in book which is exp(2*x*z-z^2)

the paper assume f(m) = 0, do not know the case when f(m) <> 0 as i do not know how to solve differential system of k0 and k1 into different type

Maple Code

restart;
P := 'P';
test2 := Diff(exp(-x^2)*(Diff(P, x)), x)+exp(-x^2)*mu*P = 0;
newTest2 := subs(P = Y(x)/exp(-x^2/2), test2);
newTest12 := expand(value(newTest2)*exp(-x^2/2));
newTest12 := collect(newTest12, Y(x));
newTest12 := collect(newTest12, Diff(Y(x),x));
newTest12 := expand(newTest12*exp(x^2));
newTest12 := collect(newTest12, Y(x));
newTest12 := collect(newTest12, Diff(Y(x),x));

R := 1-x^2;
lambda := mu;

r := -(b*x+d)^2+b*(2*m+1);
b := 1;
d := 0;

R := r + f(m);
R := -x^2+2*m+1 -2*m;
f := -2*m;

k := b*x + d;
k := x;

L := -2*b*m;
L := -2*m;

Y := C*exp(-int(x,x));
solve(int(Y^2, x=-infinity..infinity)=1,C);
solve(limit(int(Y^2,x),x=infinity)-limit(int(Y^2,x),x=-infinity)=1, C);

with(SumTools[IndefiniteSum]):
gf := Indefinite(Y*z^m/m!,m);