|
In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834 - 1886), are the canonical solutions of Laguerre's equation: 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 , are a polynomial sequence which may be defined by the Rodrigues formula L_n(x)= rac racleft(e^ x^n ight). They are orthogonal to each other with respect to the inner product given by angle = int_0^infty f(x) g(x) e^,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 , than the definition used here. The first few polynomials These are the first few Laguerre polynomials:
As contour integral The polynomials may be expressed in terms of a contour integral 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 ight. then eq 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 , ight. (see gamma function) is given by the defining Rodrigues equation for the generalized Laguerre polynomials: left(e^ x^ ight) . These are also sometimes called the associated Laguerre polynomials. The simple Laguerre polynomials are recovered from the generalized polynomials by setting α = 0: The associated Laguerre polynomials are orthogonal over with respect to the weighting function : The associated Laguerre polynomials obey the following differential equation x L_n^(x) + (alpha+1-x)L_n^(x) + n L_n^(x)=0., Explicit examples of generalized Laguerre polynomials The generalized Laguerre polynomial of degree is (as follows from applying Leibniz's theorem for differentiation of a product to the defining Rodrigues formula) L_n^ (x) = sum_^n rac from which we see that the coefficient of the leading term is and the constant term (which is also the value at the origin) is . The first few generalized Laguerre polynomials are + rac Derivatives of generalized Laguerre polynomials Differentiating the power series representation of a generalized Laguerre polynomial times leads to rac L_n^ (x) (-1)^k L_^ (x),. 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 and where the are the Hermite polynomials. Relation to hypergeometric functions The Laguerre polynomials may be defined in terms of hypergeometric functions, specifically the confluent hypergeometric functions, as where is the Pochhammer symbol (which in this case represents the rising factorial). | |||||||||||||||||||||||
|
| ||||||||||||||||||||||||
 |
|
| |