orthogonality of Laguerre polynomials

We use the definition of Laguerre polynomialsDlmfDlmfDlmfMathworldPlanetmath Ln(x) via their Rodrigues formulaPlanetmathPlanetmath (http://planetmath.org/RodriguesFormula)

Ln(x):=exdndxn(xne-x). (1)

The polynomialsPlanetmathPlanetmath (1) themselves are not orthogonal to each other, but the expressions e-x2Ln(x)  (n=0, 1, 2,) are orthogonal (http://planetmath.org/OrthogonalPolynomials) on the interval from 0 to , i.e. the polynomials are orthogonal with respect to the weighting function e-x on that interval, as is seen in the following.

Let m be another nonnegative integer.  We integrate by parts (http://planetmath.org/IntegrationByParts) m times in


When  m<n,  this yields

0e-xxmLn(x)𝑑x=(-1)mm!/x=0dn-m-1dxn-m-1(xme-x)= 0. (2)

and for  m=n  it gives

0e-xxmLn(x)𝑑x=(-1)nn!0xne-x𝑑x=(-1)n(n!)2. (3)

The result (2) implies, because Lm(x) is a polynomial of degree m, that

0e-xLm(x)Ln(x)dx= 0  (m<n),

whence also

0e-xLm(x)Ln(x)dx= 0  (mn). (4)

Thus the orthogonality has been shown.  Therefore, since the leading term of Ln(x) is (-1)nxn, we infer by (3) and (4) that


so that the expressions Ln(x)n! form a system of orthonormal polynomials.


  • 1 H. Eyring, J. Walter, G. Kimball: Quantum chemistry.  Eight printing.  Wiley & Sons, New York (1958).
Title orthogonality of Laguerre polynomials
Canonical name OrthogonalityOfLaguerrePolynomials
Date of creation 2013-03-22 19:05:50
Last modified on 2013-03-22 19:05:50
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 8
Author pahio (2872)
Entry type Derivation
Classification msc 33D45
Classification msc 33C45
Classification msc 26C05
Related topic SubstitutionNotation
Related topic PropertiesOfOrthogonalPolynomials