Laguerre polynomials
From Wikipedia, the free encyclopedia
In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834 - 1886), are the canonical solutions of Laguerre's equation:
- <math>x\,y + (1 - x)\,y' + n\,y = 0\,</math>
which is a second-order linear differential equation. This equation has nonsingular solutions only if n is a non-negative integer.
These polynomials, usually denoted <math>L_0, L_1, \dots</math>, are a polynomial sequence which may be defined by the Rodrigues formula
- <math>
L_n(x)=\frac{e^x}{n!}\frac{d^n}{dx^n}\left(e^{-x} x^n\right). </math>
They are orthogonal to each other with respect to the inner product given by
- <math>\langle f,g \rangle = \int_0^\infty f(x) g(x) e^{-x}\,dx.</math>
The sequence of Laguerre polynomials is a Sheffer sequence.
The Laguerre polynomials arise in quantum mechanics, in the radial part of the solution of the Schrödinger equation for a one-electron atom.
Physicists often use a definition for the Laguerre polynomials that is larger, by a factor of <math>(n!)</math>, than the definition used here.
Contents |
[edit] The first few polynomials
These are the first few Laguerre polynomials:
| n | <math>L_n(x)\,</math> |
| 0 | <math>1\,</math> |
| 1 | <math>-x+1\,</math> |
| 2 | <math>\begin{matrix}\frac12\end{matrix} (x^2-4x+2) \,</math> |
| 3 | <math>\begin{matrix}\frac16\end{matrix} (-x^3+9x^2-18x+6) \,</math> |
| 4 | <math>\begin{matrix}\frac1{24}\end{matrix} (x^4-16x^3+72x^2-96x+24) \,</math> |
| 5 | <math>\begin{matrix}\frac1{120}\end{matrix} (-x^5+25x^4-200x^3+600x^2-600x+120) \,</math> |
| 6 | <math>\begin{matrix}\frac1{720}\end{matrix} (x^6-36x^5+450x^4-2400x^3+5400x^2-4320x+720) \,</math> |
[edit] As contour integral
The polynomials may be expressed in terms of a contour integral
- <math>L_n(x)=\frac{1}{2\pi i}\oint\frac{e^{-xt/(1-t)}}{(1-t)\,t^{n+1}} \; dt</math>
where the contour circles the origin once in a counterclockwise direction.
[edit] Recursive definition
We can also define the Laguerre polynomials recursively, defining the first two polynomials as
- <math>L_0(x) = 1\,</math>
- <math>L_1(x) = 1 - x\,</math>
and then using the recurrence relation for any <math>k \geq 1</math>:
- <math>L_{k + 1}(x) = \frac{1}{k + 1} \bigg( (2k + 1 - x)L_k(x) - k L_{k - 1}(x)\bigg) </math>
[edit] Generalized Laguerre polynomials
The orthogonality property stated above is equivalent to saying that if X is an exponentially distributed random variable with probability density function
- <math>f(x)=\left\{\begin{matrix} e^{-x} & \mbox{if}\ x>0, \\ 0 & \mbox{if}\ x<0, \end{matrix}\right.</math>
then
- <math>E(L_n(X)L_m(X))=0\ \mbox{whenever}\ n\neq m.</math>
The exponential distribution is not the only gamma distribution. A polynomial sequence orthogonal with respect to the gamma distribution whose probability density function is, for <math>\alpha>-1</math>,
- <math>f(x)=\left\{\begin{matrix} x^\alpha e^{-x}/\Gamma(1+\alpha) & \mbox{if}\ x>0, \\ 0 & \mbox{if}\ x<0, \end{matrix}\right.</math>
(see gamma function) is given by the defining Rodrigues equation for the generalized Laguerre polynomials:
- <math>L_n^{(\alpha)}(x)=
{x^{-\alpha} e^x \over n!}{d^n \over dx^n} \left(e^{-x} x^{n+\alpha}\right) .</math>
These are also sometimes called the associated Laguerre polynomials. The simple Laguerre polynomials are recovered from the generalized polynomials by setting α = 0:
- <math>L^{(0)}_n(x)=L_n(x).</math>
The associated Laguerre polynomials are orthogonal over <math>[0,\infty)</math> with respect to the weighting function <math>x^\alpha e^{-x}</math>:
- <math>\int_0^{\infty}e^{-x}x^\alpha L_n^{(\alpha)}(x)L_m^{(\alpha)}(x)dx=\frac{\Gamma(n+\alpha+1)}{n!}\delta_{nm}.</math>
The associated Laguerre polynomials obey the following differential equation:
- <math>
x L_n^{(\alpha) \prime\prime}(x) + (\alpha+1-x)L_n^{(\alpha)\prime}(x) + n L_n^{(\alpha)}(x)=0.\, </math>
They obey the following recurrence relation for <math>n \geq 1</math>:
- <math>L_{n + 1}^\alpha(x) = \frac{1}{n + 1} \bigg( (2n + 1 + \alpha - x)L_n^\alpha(x) - (n + \alpha) L_{n - 1}^\alpha(x)\bigg).</math>
[edit] Explicit examples of generalized Laguerre polynomials
The generalized Laguerre polynomial of degree <math>n</math> is (as follows from applying Leibniz's theorem for differentiation of a product to the defining Rodrigues formula)
- <math>
L_n^{(\alpha)} (x) = \sum_{m=0}^n {n+\alpha \choose n-m} \frac{(-x)^m}{m!} </math> from which we see that the coefficient of the leading term is <math>(-1)^n/n!</math> and the constant term (which is also the value at the origin) is <math>{n+\alpha\choose n}</math>.
The first few generalized Laguerre polynomials are
- <math> L_0^{(\alpha)} (x) = 1 </math>
- <math> L_1^{(\alpha)}(x) = -x + \alpha +1</math>
- <math> L_2^{(\alpha)}(x) = \frac{x^2}{2} - (\alpha + 2)x + \frac{(\alpha+2)(\alpha+1)}{2}</math>
- <math> L_3^{(\alpha)}(x) = \frac{-x^3}{6} + \frac{(\alpha+3)x^2}{2} - \frac{(\alpha+2)(\alpha+3)x}{2}
+ \frac{(\alpha+1)(\alpha+2)(\alpha+3)}{6}</math>
[edit] Derivatives of generalized Laguerre polynomials
Differentiating the power series representation of a generalized Laguerre polynomial <math>k</math> times leads to
- <math>
\frac{\mathrm d^k}{\mathrm d x^k} L_n^{(\alpha)} (x) = (-1)^k L_{n-k}^{(\alpha+k)} (x)\,. </math>
[edit] Relation to Hermite polynomials
The generalized Laguerre polynomials are related to the Hermite polynomials:
- <math>H_{2n}(x) = (-1)^n\ 2^{2n}\ n!\ L_n^{(-1/2)} (x^2)</math>
and
- <math>H_{2n+1}(x) = (-1)^n\ 2^{2n+1}\ n!\ x\ L_n^{(1/2)} (x^2)</math>
where the <math>H_n(x)</math> are the Hermite polynomials.
Because of this, the generalized Laguerre polynomials arise in the treatment of the quantum harmonic oscillator.
[edit] Relation to hypergeometric functions
The Laguerre polynomials may be defined in terms of hypergeometric functions, specifically the confluent hypergeometric functions, as
- <math>L^{(\alpha)}_n(x) = {n+\alpha \choose n} M(-n,\alpha+1,x) =\frac{(\alpha+1)_n} {n!} \,_1F_1(-n,\alpha+1,x)</math>
where <math>(a)_n</math> is the Pochhammer symbol (which in this case represents the rising factorial).
[edit] External links
[edit] References
- Milton Abramowitz and Irene A. Stegun, eds. (1965). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover. ISBN 0-486-61272-4. (See chapter 22)
- Eric W. Weisstein, "Laguerre Polynomial", From MathWorld--A Wolfram Web Resource.
- George Arfken and Hans Weber (2000). Mathematical Methods for Physicists. Academic Press. ISBN 0-12-059825-6.de:Laguerre-Polynome
fr:Polynôme de Laguerre it:Polinomi di Laguerre nl:Laguerre-polynoom
