Laguerre differential operator on polynomials
= Laguerre differential operator on polynomials
{c}
On polynomials with weighted inner product
$$
\langle f,g\rangle=\int_0^\infty f(x)g(x)e^{-x}\,dx,
$$
the operator
$$
Lf=xf''+(1-x)f'
$$
is self-adjoint because $e^{-x}Lf=(xe^{-x}f')'$. Its eigenvalues on polynomials of degree at most $n$ are $0,-1,\ldots,-n$; its eigenvectors are scalar multiples of the <Laguerre polynomials>.