orthogonality of Laguerre polynomials

We use the definition of Laguerre polynomials $L_{n}(x)$ via their Rodrigues formula (http://planetmath.org/RodriguesFormula)

\displaystyle L_{n}(x)\;:=\;e^{x}\frac{d^{n}}{dx^{n}}(x^{n}e^{-x}).

(1)

The polynomials (1) themselves are not orthogonal to each other, but the expressions $e^{-\frac{x}{2}}L_{n}(x)$ ( $n=0,\,1,\,2,\ldots$ ) are orthogonal (http://planetmath.org/OrthogonalPolynomials) on the interval from 0 to $\infty$ , 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

\int_{0}^{\infty}\!e^{-x}x^{m}L_{n}(x)\,dx\;=\;\int_{0}^{\infty}\!x^{m}\frac{d% ^{n}}{dx^{n}}(x^{n}e^{-x})\,dx\;=\;(-1)^{m}m!\int_{0}^{\infty}\!\frac{d^{n-m}}% {dx^{n-m}}(x^{m}e^{-x})\,dx.

When $m<n$ , this yields

\displaystyle\int_{0}^{\infty}\!e^{-x}x^{m}L_{n}(x)\,dx\;=\;(-1)^{m}m!% \operatornamewithlimits{\Big{/}}_{\!\!\!x=0}^{\,\quad\infty}\!\frac{d^{n-m-1}}% {dx^{n-m-1}}(x^{m}e^{-x})\;=\;0.

(2)

and for $m=n$ it gives

\displaystyle\int_{0}^{\infty}\!e^{-x}x^{m}L_{n}(x)\,dx\;=\;(-1)^{n}n!\int_{0}% ^{\infty}\!x^{n}e^{-x}\,dx\;=\;(-1)^{n}(n!)^{2}.

(3)

The result (2) implies, because $L_{m}(x)$ is a polynomial of degree $m$ , that

\int_{0}^{\infty}\!e^{-x}L_{m}(x)L_{n}(x)\,dx\;=\;0\qquad(m\;<\;n),

whence also

\displaystyle\int_{0}^{\infty}\!e^{-x}L_{m}(x)L_{n}(x)\,dx\;=\;0\qquad(m\;\neq% \;n).

(4)

Thus the orthogonality has been shown. Therefore, since the leading term of $L_{n}(x)$ is $(-1)^{n}x^{n}$ , we infer by (3) and (4) that

\int_{0}^{\infty}\!e^{-x}[L_{n}(x)]^{2}\,dx\;=\;(-1)^{n}\!\int_{0}^{\infty}\!e% ^{-x}x^{n}L_{n}(x)\,ds\;=\;(n!)^{2},

so that the expressions $\frac{L_{n}(x)}{n!}$ form a system of orthonormal polynomials.

References

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