= Zero-eigenvalue solutions of the Laguerre differential equation
{title2=$y=C+D\operatorname{Ei}(x),\quad x>0$}
At eigenvalue zero the <Laguerre differential operator on polynomials> gives $xy''+(1-x)y'=0$, equivalently $(xe^{-x}y')'=0$. Thus $y'=De^x/x$ and the displayed solutions follow from the <exponential integral>. For any $x_0>0$, all solutions with $y(x_0)=1$ can be written $y(x)=1+D\int_{x_0}^xe^t\,dt/t$. The choices $D=0$ and $D=1$ are linearly independent despite sharing that one prescribed value.
The second branch grows as $e^x/x$ at infinity and has a logarithmic singularity at zero. It is consequently excluded from the regular weighted square-integrable eigenproblem with weight $e^{-x}$. Only the constant zero mode remains in that self-adjoint realization. A normalization at one interior point alone is not enough to select a unique second-order solution.
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