Question: find generating function of Hermit

The goal is to find generating function of Hermit

Before i try, i try the gFun's paper's example, got error

with(gfun):
eq := {z^2*diff(y(z),z)+y(z)=z, (D(y))(0)=1};
eq2:=borel(eq,y(z),Y,'diffeq');
Error, (in gfun:-borel) invalid argument, Y

When try Hermit,

with(gfun):
eq := {t*diff(y(z),z$2)-z*diff(y(z),z)+y(z)=0,y(0)=1,y(-z)=y(z)};
dsolve(eq);

but have unknown constant coefficient, do not know whether missing initial condition

and try

eq := {t*diff(y(z),z$2)-z*diff(y(z),z)+y(z)=0,(D(y))(0)=1};
dsolve(eq);

2. Do not know which kind of differential equation is suitable for generating function, then try this,
got error, it is not linear differential equations
with(gfun):
opoly := ((z*diff(f(z),z))^5)-3*z*(2*(z*diff(f(z),z))+1)*(3*((z*diff(f(z),z)))^2+3*(z*diff(f(z),z))+1)*(15*(z*diff(f(z),z))^2+15*(z*diff(f(z),z))+4)-3*(z^2)*(((z*diff(f(z),z))+1)^3)*(3*(z*diff(f(z),z))+2)*(3*(z*diff(f(z),z))+4);
Recur := diffeqtorec(opoly,f(z), Q(n));
erec:=borel(Recur,Q(n));
edeq:=rectodiffeq(erec,Q(n),y(z)):
dsolve(edeq,y(z));

with(gfun):
opoly := t*diff(f(z),z$2)-z*diff(f(z),z)+f(z)=0;
Recur := diffeqtorec(opoly,f(z), Q(n));
erec:=borel(Recur,Q(n));
edeq:=rectodiffeq({erec,Q(0)=1,Q(-z)=Q(z)},Q(n),y(z)):
dsolve(edeq,y(z));