orthogonality of Laguerre polynomials

 $\displaystyle L_{n}(x)\;:=\;e^{x}\frac{d^{n}}{dx^{n}}(x^{n}e^{-x}).$ (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,  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 OrthogonalityOfLaguerrePolynomials 2013-03-22 19:05:50 2013-03-22 19:05:50 pahio (2872) pahio (2872) 8 pahio (2872) Derivation msc 33D45 msc 33C45 msc 26C05 SubstitutionNotation PropertiesOfOrthogonalPolynomials