= Solution
Normalize the <Laguerre differential equation> as $y''+P(x)y'+Q(x)y=0$, where $P(x)=1/x-1$ and $Q(x)=n/x$. At $x=0$, the products $xP=1-x$ and $x^2Q=nx$ are <analytic function>[analytic]. Thus zero is a <regular singular point>, with <indicial equation>
$$
\boxed{r(r-1)+r=r^2=0,\qquad r_1=r_2=0.}
$$
At $x=1$ the coefficients $P,Q$ are <analytic function>[analytic], so it is an \b[<ordinary point>]. No singular-point <indicial equation> is needed there.
For infinity put $z=1/x$ and $Y(z)=y(1/z)$. Then the equation becomes
$$
Y''+(z^{-1}+z^{-2})Y'+nz^{-3}Y=0.
$$
At $z=0$, the coefficient $z(z^{-1}+z^{-2})=1+1/z$ is not <analytic function>[analytic]. Therefore infinity is an \b[<irregular singular point>], including when $n=0$. A terminating <polynomial> solution does not change the classification of the differential equation.
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