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The sequence of Laguerre polynomials is a Sheffer sequence.
The first few polynomials are
- L0(x) = 1
- L1(x) = - x + 1
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 exponential distribution is not the only gamma distribution. A polynomial sequence orthogonal with respect to the gamma distribution whose probability density function is
(see gamma function) is given by the defining equation for the generalized Laguerre polynomials:
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 xαe - x:
For integer α the defining equation above can be written as
Relation to Hermite polynomials
where the Hn(x) are the Hermite polynomials.
Relation to hypergeometric functions
where (a)n is the Pochhammer symbol.
- Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions, (1964) Dover Publications, New York. ISBN 486-61272-4 . (See chapter 22).
- Laguerre polynomial on MathWorld
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