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

$x\backslash ,y\text{'}\text{'}\; +\; (1\; -\; x)\backslash ,y\text{'}\; +\; n\backslash ,y\; =\; 0\backslash ,$

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 $L\_0,\; L\_1,\; \backslash dots$, 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

$\backslash langle\; f,g\; \backslash rangle\; =\; \backslash int\_0^\backslash infty\; f(x)\; g(x)\; e^\{-x\}\backslash ,dx.$

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.

The first few polynomials

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These are the first few Laguerre polynomials:

n	$L\_n(x)\backslash ,$
0	$1\backslash ,$
1	$-x+1\backslash ,$
2	$\backslash begin\{matrix\}\backslash frac12\backslash end\{matrix\}\; (x^2-4x+2)\; \backslash ,$
3	$\backslash begin\{matrix\}\backslash frac16\backslash end\{matrix\}\; (-x^3+9x^2-18x+6)\; \backslash ,$
4	$\backslash begin\{matrix\}\backslash frac1\{24\}\backslash end\{matrix\}\; (x^4-16x^3+72x^2-96x+24)\; \backslash ,$
5	$\backslash begin\{matrix\}\backslash frac1\{120\}\backslash end\{matrix\}\; (-x^5+25x^4-200x^3+600x^2-600x+120)\; \backslash ,$
6	$\backslash begin\{matrix\}\backslash frac1\{720\}\backslash end\{matrix\}\; (x^6-36x^5+450x^4-2400x^3+5400x^2-4320x+720)\; \backslash ,$

As contour integral

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The polynomials may be expressed in terms of a contour integral

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

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

Generalized Laguerre polynomials

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The orthogonality property stated above is equivalent to saying that if X is an exponentially distributed random variable with probability density function

then

$E(L\_n(X)L\_m(X))=0\backslash \; \backslash mbox\{whenever\}\backslash \; n\backslash neq\; m.$

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 $\backslash alpha>-1$,

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

$L\_n^\{(\backslash alpha)\}(x)=$

{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 $[0,\backslash infty)$ with respect to the weighting function $x^\backslash alpha\; e^\{-x\}$:

$\backslash int\_0^\{\backslash infty\}e^\{-x\}x^\backslash alpha\; L\_n^\{(\backslash alpha)\}(x)L\_m^\{(\backslash alpha)\}(x)dx=\backslash frac\{\backslash Gamma(n+\backslash alpha+1)\}\{n!\}\backslash delta\_\{nm\}.$

The associated Laguerre polynomials obey the following differential equation

$$

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

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_{m=0}^n {n+\alpha \choose n-m} \frac{(-x)^m}{m!} 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 $\{n+\backslash alpha\backslash choose\; n\}$.

The first few generalized Laguerre polynomials are

$L\_0^\{(\backslash alpha)\}\; (x)\; =\; 1$

$L\_1^\{(\backslash alpha)\}(x)\; =\; -x\; +\; \backslash alpha\; +1$

$L\_2^\{(\backslash alpha)\}(x)\; =\; \backslash frac\{x^2\}\{2\}\; -\; (\backslash alpha\; +\; 2)x\; +\; \backslash frac\{(\backslash alpha+2)(\backslash alpha+1)\}\{2\}$

$L\_3^\{(\backslash alpha)\}(x)\; =\; \backslash frac\{-x^3\}\{6\}\; +\; \backslash frac\{(\backslash alpha+3)x^2\}\{2\}\; -\; \backslash frac\{(\backslash alpha+2)(\backslash alpha+3)x\}\{2\}$

+ \frac{(\alpha+1)(\alpha+2)(\alpha+3)}{6}

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)\,.

Relation to Hermite polynomials

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The generalized Laguerre polynomials arise in the treatment of the quantum harmonic oscillator, due to their relation to the Hermite polynomials, which can be expressed as

$H\_\{2n\}(x)\; =\; (-1)^n\; 2^\{2n\}\; n!\; L\_n^\{(-1/2)\}\; (x^2)$

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.

Relation to hypergeometric functions

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

$L^\{(\backslash alpha)\}\_n(x)\; =\; \{n+\backslash alpha\; \backslash choose\; n\}\; M(-n,\backslash alpha+1,x)\; =\backslash frac\{(\backslash alpha+1)\_n\}\; \{n!\}\; \backslash ,\_1F\_1(-n,\backslash alpha+1,x)$

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

External links

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

References

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• (See chapter 22.)
• Eric W. Weisstein, "Laguerre Polynomial", From MathWorld--A Wolfram Web Resource.

Orthogonal polynomials | Special hypergeometric functions

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