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

seriestoratpoly(series(simplify(LaguerreL(5, x)),x,6),['egf']);

seriestoratpoly(series(simplify(LaguerreL(5, a, x)),x,6),['egf']);

inverse z transform of sqrt(1-a^2)/(1-a/z)

restart;
alpha := 0;
s := x;
tau := alpha + 1 - x;
source := s*Diff(y(x),x$2)+tau*Diff(y(x),x)+n*y(x)=0;
sol := dsolve(source,y(x),method=laplace);
zsol := ztrans(sol,x,z);
convert(zsol, ratpoly);

sol := y(x) = _C1*invlaplace((_s1-1)^n*_s1^(-1-n), _s1, x)
zsol := ztrans(y(x), x, z) = _C1*invlaplace((_s1-1)^n*_s1^(-1-n)*z/(z-1), _s1, x)
Error, (in convert/ratpoly) expecting a Laurent series or a Chebyshev series

what are _C1 and _s1?

except kummer representation, can dsolve output simple? or any other method to do this?

s := x;
tau := alpha - 1 - x;
source := s*Diff(y(x),x$2)+tau*Diff(y(x),x)+n*y(x)=0;
sol := dsolve(source,y(x));
target1 :=ztrans(sol,x,z);

when N = 1
target2 := sqrt(1-a^2)/(1-a/z)*((1/z-a)/(1-a/z))^(N-1);

Gamma := sqrt(1-a^2)/(1-a/z)*((1/z-a)/(1-a/z))^(N-1);
int(subs(z=exp(j*w),Gamma)*Conjugate(subs(z=exp(j*w),Gamma)),w=-Pi..Pi)

would like to prove this integration = 2*Pi, however got error

int(subs(z=exp(j*w),Gamma)*subs(z=exp(-j*w),Gamma),w=-Pi..Pi) also not equal to 2*Pi