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Laguerre differential operator on polynomials

On polynomials with weighted inner product

⟨ f, g ⟩ = \int_{0}^{\infty} f (x) g (x) e^{- x} d x,

the operator

L f = x f^{''} + (1 - x) f^{'}

is self-adjoint because

e^{- x} L f = (x e^{- x} f^{'})^{'}

. Its eigenvalues on polynomials of degree at most

n

are

0, - 1, \dots, - n

; its eigenvectors are scalar multiples of the Laguerre polynomials.

Table of contents 76 3
- Laguerre polynomial Laguerre differential operator on polynomials 39 2
  - Generalized Laguerre polynomial Laguerre polynomial 39 1
    - Rodrigues formula Generalized Laguerre polynomial 22

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