Laguerre polynomials

Laguerre polynomials
This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Laguerre_polynomials".

In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834 - 1886), are the canonical solutions of Laguerre's equation:

$x\,y'' + (1 - x)\,y' + n\,y = 0\,$

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 $L_0, L_1, \dots$ , are a polynomial sequence which may be defined by the Rodrigues formula

$L_n(x)=\frac{e^x}{n!}\frac{d^n}{dx^n}\left(e^{-x} x^n\right).$

They are orthogonal to each other with respect to the inner product given by

$\langle f,g \rangle = \int_0^\infty f(x) g(x) e^{-x}\,dx.$

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 $\, (n!)$ , than the definition used here.

content

1 The first few polynomials
2 Recursive definition
3 Generalized Laguerre polynomials
4 As contour integral
5 Relation to Hermite polynomials
6 Relation to hypergeometric functions
7 Computation using Horner's method
8 External links
9 References

The first few polynomials

These are the first few Laguerre polynomials:

n	$L_n(x)\,$
0	$1\,$
1	$-x+1\,$
2	${\scriptstyle\frac{1}{2}} (x^2-4x+2) \,$
3	${\scriptstyle\frac{1}{6}} (-x^3+9x^2-18x+6) \,$
4	${\scriptstyle\frac{1}{24}} (x^4-16x^3+72x^2-96x+24) \,$
5	${\scriptstyle\frac{1}{120}} (-x^5+25x^4-200x^3+600x^2-600x+120) \,$
6	${\scriptstyle\frac{1}{720}} (x^6-36x^5+450x^4-2400x^3+5400x^2-4320x+720) \,$

The first six Laguerre polynomials.

Recursive definition

We can also define the Laguerre polynomials recursively, defining the first two polynomials as

$L_0(x) = 1\,$

$L_1(x) = 1 - x\,$

and then using the recurrence relation for any $k \geq 1$ :

$L_{k + 1}(x) = \frac{1}{k + 1} \bigg( (2k + 1 - x)L_k(x) - k L_{k - 1}(x)\bigg).$

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

$f(x)=\left\{\begin{matrix} e^{-x} & \mbox{if}\ x>0, \\ 0 & \mbox{if}\ x<0, \end{matrix}\right.$

then

$E \left[ L_n(X)L_m(X) \right]=0\ \mbox{whenever}\ n\neq m.$

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 $α > - 1$ ,

$f(x)=\left\{\begin{matrix} x^\alpha e^{-x}/\Gamma(1+\alpha) & \mbox{if}\ x>0, \\ 0 & \mbox{if}\ x<0, \end{matrix}\right.$

(see gamma function) is given by the defining Rodrigues equation for the generalized Laguerre polynomials:

$L_n^{(\alpha)}(x)= {x^{-\alpha} e^x \over n!}{d^n \over dx^n} \left(e^{-x} x^{n+\alpha}\right) .$

These are also sometimes called the associated Laguerre polynomials. The simple Laguerre polynomials are recovered from the generalized polynomials by setting α = 0:

$L^{(0)}_n(x)=L_n(x).$

Explicit examples of generalized Laguerre polynomials

The generalized Laguerre polynomial of degree $n$ is (as follows from applying Leibniz's theorem for differentiation of a product to the defining Rodrigues formula)

$L_n^{(\alpha)} (x) = \sum_{i=0}^n (-1)^i {n+\alpha \choose n-i} \frac{x^i}{i!}$

from which we see that the coefficient of the leading term is $( - 1) n / n!$ and the constant term (which is also the value at the origin) is ${n+\alpha\choose n}.$

The first few generalized Laguerre polynomials are:

$L_0^{(\alpha)} (x) = 1$

$L_1^{(\alpha)}(x) = -x + \alpha +1$

$L_2^{(\alpha)}(x) = \frac{x^2}{2} - (\alpha + 2)x + \frac{(\alpha+2)(\alpha+1)}{2}$

$L_3^{(\alpha)}(x) = \frac{-x^3}{6} + \frac{(\alpha+3)x^2}{2} - \frac{(\alpha+2)(\alpha+3)x}{2} + \frac{(\alpha+1)(\alpha+2)(\alpha+3)}{6}$

Recurrence Relations

Laguerre's polynomials satisfy the recurrence relations

$L_n^{(\alpha+\beta+1)}(x+y)= \sum_{i=0}^n L_i^{(\alpha)}(x) L_{n-i}^{(\beta)}(y),$

in particular

$L_n^{(\alpha+1)}(x)= \sum_{i=0}^n L_i^{(\alpha)}(x);$ moreover

$L_n^{(\alpha)}(x)= {n+\alpha \choose n} - \frac{x}{n} \sum_{i=0}^{n-1} \frac{{n+\alpha \choose n-1-i}}{{n-1 \choose i}}L_i^{(\alpha+1)}(x).$

They can be used to derive

$L_n^{(\alpha)}(x) = L_n^{(\alpha+1)}(x) - L_{n-1}^{(\alpha+1)}(x)$ and

$n L_n^{(\alpha)}(x) = (n + \alpha )L_{n-1}^{(\alpha)}(x) - x L_{n-1}^{(\alpha+1)}(x);$

combined they give this additional, popular recurrence relation

$L_{n + 1}^{(\alpha)}(x) = \frac{1}{n + 1} \bigg( (2n + 1 + \alpha - x)L_n^{(\alpha)}(x) - (n + \alpha) L_{n - 1}^{(\alpha)}(x)\bigg).$

Derivatives of generalized Laguerre polynomials

Differentiating the power series representation of a generalized Laguerre polynomial $k$ times leads to

$\frac{\mathrm d^k}{\mathrm d x^k} L_n^{(\alpha)} (x) = (-1)^k L_{n-k}^{(\alpha+k)} (x)\,;$

moreover, this following equation holds

$\frac{1}{k!} \frac{\mathrm d^k}{\mathrm d x^k} x^\alpha L_n^{(\alpha)} (x) = {n+\alpha \choose k} x^{\alpha-k} L_n^{(\alpha-k)}(x) \,.$

The generalized associated Laguerre polynomials obey the differential equation

$x L_n^{(\alpha) \prime\prime}(x) + (\alpha+1-x)L_n^{(\alpha)\prime}(x) + n L_n^{(\alpha)}(x)=0.\,$

Orthogonality

The associated Laguerre polynomials are orthogonal over $[0,\infty)$ with respect to the measure with weighting function $x α e - x$ :

$\int_0^{\infty}x^\alpha e^{-x} L_n^{(\alpha)}(x)L_m^{(\alpha)}(x)dx=\frac{\Gamma(n+\alpha+1)}{n!}\delta_{n,m}.$

The associated, symmetric kernel polynomial has the representations

$\begin{align} K_n^{(\alpha)}(x,y)&{:=}\frac{1}{\Gamma(\alpha+1)} \sum_{i=0}^n \frac{L_i^{(\alpha)}(x) L_i^{(\alpha)}(y)}{{\alpha+i \choose i}}\\ &{=}\frac{1}{\Gamma(\alpha+1)} \frac{L_n^{(\alpha)}(x) L_{n+1}^{(\alpha)}(y) - L_{n+1}^{(\alpha)}(x) L_n^{(\alpha)}(y)}{\frac{x-y}{n+1} {n+\alpha \choose n}} \\ &{=}\frac{1}{\Gamma(\alpha+1)}\sum_{i=0}^n \frac{x^i}{i!} \frac{L_{n-i}^{(\alpha+i)}(x) L_{n-i}^{(\alpha+i+1)}(y)}{{\alpha+n \choose n}{n \choose i}};\end{align}$

recursively

$K_n^{(\alpha)}(x,y)=\frac{y}{\alpha+1} K_{n-1}^{(\alpha+1)}(x,y)+ \frac{1}{\Gamma(\alpha+1)} \frac{L_n^{(\alpha+1)}(x) L_n^{(\alpha)}(y)}{{\alpha+n \choose n}}.$

Moreover, $y^\alpha e^{-y} K_n^{(\alpha)}(\, . ,y) \rightarrow \delta(y- \, .)$ , in the associated $L^2[0, \infty)$ -space.

The following integral is needed in the quantum mechanical treatment of the hydrogen atom,

$\int_0^{\infty}x^{\alpha+1} e^{-x} \left[L_n^{(\alpha)}\right]^2 dx= \frac{(n+\alpha)!}{n!}(2n+\alpha+1).$

Series Expansions

Monomials are representated as

$\frac{x^n}{n!}= \sum_{i=0}^n (-1)^i {n+ \alpha \choose n-i} L_i^{(\alpha)}(x).$

This leads directly to

$e^{-\gamma x}= \sum_{i=0} \frac{\gamma^i}{(1+\gamma)^{i+\alpha+1}} L_i^{(\alpha)}(x)$

and, even more generally,

$\frac{x^\beta e^{-\gamma x}}{\Gamma(\beta+1)}= {\alpha+\beta \choose \alpha} \sum_{i=0} \frac{L_i^{(\alpha)}(x)}{ {\alpha+i \choose i}} \sum_{j=0}^i \frac{(-1)^j}{(1+\gamma)^{\alpha+ \beta+ j+ 1}} {\alpha+\beta+j \choose j} {\alpha+i \choose i-j}.$

For $β$ a non-negative integer this simplifies to

$\frac{x^n e^{-\gamma x}}{n!}= \sum_{i=0} \frac{\gamma^i L_i^{(\alpha)}(x)}{(1+\gamma)^{i+n+\alpha+1}} \sum_{j=0}^n (-1)^{n-j} \gamma^j {n+\alpha \choose j} {i \choose n-j},$

for $γ = 0$ to

$\frac{x^\beta}{\Gamma(\beta+1)} = {\alpha+ \beta \choose \alpha} \sum_{i=0} (-1)^i {\beta \choose i} \frac{L_i^{(\alpha)}(x)}{{\alpha+i \choose i}}.$

The Bessel function $J α$ can be expressed (using an arbitrarily chosen parameter $t$ ) as

$\frac{J_\alpha(x)}{\left( \frac{x}{2}\right)^\alpha}= \frac{e^{-t}}{\Gamma(\alpha+1)} \sum_{i=0} \frac{L_i^{(\alpha)}\left( \frac{x^2}{4 t}\right)}{{i+ \alpha \choose i}} \frac{t^i}{i!}.$

This Taylor series expansion, somewhat more comprehensive,

$\frac{e^{-\frac{x t}{1-t}}}{(1-t)^{\alpha+1}} = \sum_{i=0} t^i L_i^{(\alpha)}(x)$

is an immediate consequence of the definition of Laguerre's polynomials; derived from that are the identities

$\frac{e^{-\frac{x t}{1-t}}}{(1-t)^{n+a+1}}L_n^{(\alpha)}\left( \frac{x}{1-t} \right)= \sum_{i=n} {i \choose n} t^{i-n} L_i^{(\alpha)} (x)$ .

As contour integral

The polynomials may be expressed in terms of a contour integral

$L_n^{(\alpha)}(x)=\frac{1}{2\pi i}\oint\frac{e^{-\frac{x t}{1-t}}}{(1-t)^{\alpha+1}\,t^{n+1}} \; dt$

where the contour circles the origin once in a counterclockwise direction.

Relation to Hermite polynomials

The generalized Laguerre polynomials are related to the Hermite polynomials:

$H_{2n}(x) = (-1)^n\ 2^{2n}\ n!\ L_n^{(-1/2)} (x^2)$

and

$H_{2n+1}(x) = (-1)^n\ 2^{2n+1}\ n!\ x\ L_n^{(1/2)} (x^2)$

where the $H n (x)$ are the Hermite polynomials based on the weighting function $exp( - x 2)$ , the so-called "physicist's version."

Because of this, the generalized Laguerre polynomials arise in the treatment of the quantum harmonic oscillator.

Relation to hypergeometric functions

The Laguerre polynomials may be defined in terms of hypergeometric functions, specifically the confluent hypergeometric functions, as

$L^{(\alpha)}_n(x) = {n+\alpha \choose n} M(-n,\alpha+1,x) =\frac{(\alpha+1)_n} {n!} \,_1F_1(-n,\alpha+1,x)$

where $(a) n$ is the Pochhammer symbol (which in this case represents the rising factorial).

Computation using Horner's method

 function LaguerreL(n, alpha, x) {
    LaguerreL:= 1; bin:= 1 
    for i:= n to 1 step -1 {
        bin:= bin* (alpha+ i)/ (n+ 1- i)
        LaguerreL:= bin- x* LaguerreL/ i
    }
    return LaguerreL;
 }

External links

A quick informal derivation of the Laguerre polynomial in the context of the quantum mechanics of hydrogen

References

Abramowitz, Milton & Stegun, Irene A., eds. (1965), "Chapter 22", Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover, ISBN 0-486-61272-4 .
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.

Your Ad Here

Contents