Laguerre polynomials

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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.

Contents

[edit] 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.
The first six Laguerre polynomials.

[edit] As contour integral

The polynomials may be expressed in terms of a contour integral

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

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

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).

[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

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).

The associated Laguerre polynomials are orthogonal over [0,\infty) with respect to the weighting function xαe − x:

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

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

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


[edit] Recurrence Relations

Laguerre's polynomials satisfy the recurrence relations

L_n^{(\alpha+1)}(x)= \sum_{i=0}^n L_i^{(\alpha)}(x) and
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).

Moreover, the 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.\,


[edit] Series Expansions

The Taylor series expansion

\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)

and this surprising fourier series expansion of \frac{1}{x^\beta},

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

which involves the Gamma function.

Other useful expansions are

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

and, even more generally,

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

[edit] 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}

[edit] 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) \,.

[edit] 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 Hn(x) are the Hermite polynomials based on the weighting function exp( − x2), the so-called "physicist's version."

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

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).

[edit] External links

[edit] References