NationMaster - Encyclopedia: Laguerre polynomials

In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834 - 1886), are a polynomial sequence defined by Euclid, detail from The School of Athens by Raphael. ... Edmond Nicolas Laguerre (April 9, 1834 - August 14, 1886) was French mathematician who was born in Bar-le-Duc France and died in Bar-le-Duc France. ... In mathematics, a polynomial sequence is a sequence of polynomials indexed by the nonnegative integers 0, 1, 2, 3, ..., in which each index is equal to the degree of the corresponding polynomial. ...

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

These polynomials are orthogonal to each other with respect to the inner product given by In mathematics, two polynomials f and g are orthogonal to each other with respect to a nonnegative weight function w precisely if In other words, if polynomials are treated as vectors and the inner product of two polynomials f(x) and g(x) is defined as then the orthogonal polynomials... // Definition Inner Product of two vectors Given twoN-by-1 column vectors v and u, the inner product is defined as the scalar quantity Î± resulting from where or equivalently indicates the conjugate transpose operator applied to vector v. ...

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

Also, for each n, L_n(x) is a solution of Laguerre's equation

$(x D^2 + (1 - x) D + n) , y(x) = 0$

which is a second-order linear differential equation with variable coefficients. In mathematics, a linear differential equation is a differential equation Lf = g, where the differential operator L is a linear operator. ...

The sequence of Laguerre polynomials is a Sheffer sequence. In mathematics, a polynomial sequence, i. ...

1 Low orders
2 As contour integral
3 Generalized Laguerre polynomials
4 Low-order examples of generalized Laguerre polynomials
5 Relation to Hermite polynomials
6 Relation to hypergeometric functions
7 External links
8 References

Low orders

The first 5 Laguerre polynomials.

The first few polynomials are: Image File history File links Laguerre_poly. ... Image File history File links Laguerre_poly. ...

$L_0(x)=1,$

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

$L_2(x)=frac{1}{2}(x^2-4x+2)$

$L_3(x)=frac{1}{6}(-x^3+9x^2-18x+6)$

$L_4(x)=frac{1}{24}(x^4-16x^3+72x^2-96x+24)$

$L_5(x)=frac{1}{120}(-x^5+25x^4-200x^3+600x^2-600x+120)$

$L_6(x)=frac{1}{720}(x^6-36x^5+450x^4-2400x^3+5400x^2-4320x+720)$

As contour integral

The polynomials may be expressed in terms of a contour integral This article is about path integrals in the general mathematical sense, and not the path integral formulation of physics which was studied by Richard Feynman. ...

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

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

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 In probability theory and statistics, the exponential distributions are a class of continuous probability distribution. ... A random variable is a term used in mathematics and statistics. ... In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ...

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

then

$E(L_n(X)L_m(X))=0 mbox{whenever} nneq m.$

The exponential distribution is not the only gamma distribution. A polynomial sequence orthogonal with respect to the gamma distribution whose probability density function is In probability theory and statistics, the gamma distribution is a continuous probability distribution. ...

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

(see gamma function) is given by the defining equation for the generalized Laguerre polynomials: The Gamma function along part of the real axis In mathematics, the Gamma function extends the factorial function to complex and non integer numbers (it is already defined on the naturals, and has simple poles at the negative integers). ...

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

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 $x α e - x$ :

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

For integer α the defining equation above can be written as

$L_n^{(m)}(x)= (-1)^m{d^m over dx^m} L_{n+m}(x).$

The associated Laguerre polynomials obey the following differential equation

$xL_n^{(m) primeprime}(x) + (m+1-x)L_n^{(m)prime}(x) + nL_n^{(m)}(x)=0.,$

Low-order examples of generalized Laguerre polynomials

$L_0^alpha(x) = 1$

$L_1^alpha(x) = -x + alpha +1$

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

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

Relation to Hermite polynomials

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 The quantum harmonic oscillator is the quantum mechanical analogue of the classical harmonic oscillator. ... In mathematics, the Hermite polynomials, named in honor of Charles Hermite (pronounced air MEET), are a polynomial sequence defined either by (the probabilists Hermite polynomials), or sometimes by (the physicists Hermite polynomials). These two definitions are not exactly equivalent; either is a trivial rescaling of the other. ...

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

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. In mathematics, the Hermite polynomials, named in honor of Charles Hermite (pronounced air MEET), are a polynomial sequence defined either by (the probabilists Hermite polynomials), or sometimes by (the physicists Hermite polynomials). These two definitions are not exactly equivalent; either is a trivial rescaling of the other. ...

Relation to hypergeometric functions

The Laguerre polynomials may be defined in terms of hypergeometric functions, specifically the confluent hypergeometric functions, as In mathematics, a hypergeometric series could in principle be any formal power series in which the ratio of successive coefficients an/an-1 is a rational function of n. ... In mathematics, the confluent hypergeometric function is formed from hypergeometric series. ...

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

where $(a) n$ is the Pochhammer symbol (which in this case represents the rising factorial). In mathematics, the Pochhammer symbol, introduced by Leo August Pochhammer, is used in the theory of special functions to represent the rising factorial or upperfactorial and, confusingly, is used in combinatorics to represent the falling factorial or lower factorial To distinguish the two, the notations and are commonly used to...

External links

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

References

Milton Abramowitz and Irene A. Stegun, eds. (1972). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover. ISBN 0486612724. (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 0120598256.

Categories: Orthogonal polynomials | Special hypergeometric functions Abramowitz and Stegun is the informal moniker of a mathematical reference work edited by Milton Abramowitz and Irene Stegun of the U.S. National Bureau of Standards. ...

Results from FactBites:

PlanetMath: Laguerre polynomial (323 words)
	The ordinary Laguerre polynomials are the special case of the generalized Laguerre polynomials when the parameter goes to zero.
	When some result holds for generalized Laguerre polynomials which is not more complicated than that for ordinary Laguerre polynomials, we shall only provide the more general result and leave it to the reader to send the parameter to zero to recover the more specific result.
	This is version 17 of Laguerre polynomial, born on 2002-03-06, modified 2007-03-03.

Laguerre polynomials - definition of Laguerre polynomials in Encyclopedia (160 words)
	A polynomial sequence orthogonal with respect to the gamma distribution whose probabity density function is
	The sequence of Laguerre polynomials is a Sheffer sequence.
	The Laguerre polynomials are defined in terms of confluent hypergeometric functions as:

More results at FactBites »

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