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)...

After a few steps and derived out a formula include a hypergeometric expression

hypergeom(....)

in this hypergeometric expression, there are more than two parameters in maple for example

hypergeom(X,Y,Z...)

from some paper said G(z*q; q^f)

i am not familiar with hypergeometric series

how to substitute z*q and q^f into a formula include a hypergeometric expression.

ode1 := Diff(y(x),x$2) + (lambda-x^2)*y(x) = 0;

After put y = exp(-x^2/2)*y1(x) into above ode1

would like to find the solution of ode in terms of integration

in this example, it use z^(-lambda/2-1/2) = expa(-lambda/2-1/2)*log(z)

the example solution should be int(exp(...),z)

linearsol(ode1, y(x)); but return {}  nothing

sol := dsolve(t*diff(phi(x),x$2)-x*diff(phi(x),x)+n*phi(x),phi(x));
phi := unapply(rhs(sol), x);
BC := [phi(0)=0,phi(1)=0];
with(linalg):
Ccoef := genmatrix(BC, [_C1,_C2]);
CharEqn := det(Ccoef) = 0;

restart;
sol := dsolve(t*diff(phi(x,t,n),x$2)-x*diff(phi(x,t,n),x)+n*phi(x,t,n),phi(x,t,n));
phi := unapply(rhs(sol), x);
BC := [phi(0,0,0)=0,phi(1,1,1)=0];
with(linalg):
Ccoef := genmatrix(BC, [_C1,_C2]);
CharEqn := det(Ccoef) = 0;

with(SumTools):
He := sum(((-1)^k)*n!/(k!*(n-2*k)!*2^k)*x^(n-2*k), k=0..n/2);
Gen := simplify(sum(He*z^n/n!, n=0..infinity));

convert(Gen, ratpoly);