Laguerre polynomials

Chebyshev–Laguerre polynomials

Polynomials that are orthogonal on the interval $ ( 0 , \infty ) $ with weight function $ \phi ( x) = x ^ \alpha e ^ {-x}$, where $ \alpha > - 1 $. The standardized Laguerre polynomials are defined by the formula

$$ L _ {n} ^ \alpha ( x) = \ \frac{x ^ {- \alpha } e ^ {x} }{n!} \frac{d ^ {n} }{dx ^ {n} } ( x ^ {\alpha + n } e ^ {-x} ) ,\ \ n = 0 , 1 , . . . . $$

Their representation by means of the gamma-function is

$$ L _ {n} ^ \alpha ( x) = \ \sum _ { k= 0}^{ n } \frac{\Gamma ( \alpha + n + 1 ) }{\Gamma ( \alpha + k + 1 ) } \frac{( - x ) ^ {k} }{k ! ( n - k ) ! } . $$

In applications the most important formulas are:

$$ ( n + 1 ) L _ {n+1} ^ \alpha ( x) = \ ( \alpha + 2n + 1 - x ) L _ {n} ^ \alpha ( x) - ( \alpha + n ) L _ {n-1} ^ \alpha ( x) , $$

$$ x L _ {n-1} ^ {\alpha + 1 } ( x) = ( n + \alpha ) L _ {n-1} ^ \alpha ( x) - n L _ {n} ^ \alpha ( x) , $$

$$ ( L _ {n} ^ \alpha ( x) ) ^ \prime = - L _ {n-1} ^ {\alpha + 1 } ( x) . $$

The polynomial $ L _ {n} ^ \alpha ( x) $ satisfies the differential equation (Laguerre equation)

$$ x y ^ {\prime\prime} + ( \alpha - x + 1 ) y ^ \prime + n y = 0 ,\ n = 1 , 2 , . . . . $$

The generating function of the Laguerre polynomials has the form

$$ \frac{e ^ {- x t / ( 1 - t ) } }{( 1 - t ) ^ {\alpha + 1 } } = \ \sum _ { n=0}^\infty L _ {n} ^ \alpha ( x) t ^ {n} . $$

The orthonormal Laguerre polynomials can be expressed in terms of the standardized polynomials as follows:

$$ \widehat{L} {} _ {n} ^ \alpha ( x) = (- 1) ^ {n} L _ {n} ^ \alpha ( x) \sqrt { \frac{\Gamma ( n + 1 ) }{\Gamma ( \alpha + n + 1 ) } } . $$

The set of all Laguerre polynomials is dense in the space of functions whose square is integrable with weight $ \phi ( x) $ on the interval $ ( 0 , \infty ) $.

Laguerre polynomials are most frequently used under the condition $ \alpha = 0 $; these were investigated by E. Laguerre [1], and are denoted in this case by $ L _ {n} ( x) $( in contrast to them, the $ L _ {n} ^ \alpha ( x) $ are sometimes known as generalized Laguerre polynomials). The first few Laguerre polynomials $ L _ {n} ( x) $ have the form

$$ L _ {0} ( x) = 1 ,\ L _ {1} ( x) = 1 - x , $$

$$ L _ {2} ( x) = 1 - 2 x + \frac{x ^ {2} }{2} , $$

$$ L _ {3} ( x) = 1 - 3 x + \frac{3 x ^ {2} }{2} - \frac{x ^ {3} }{6} , $$

$$ L _ {4} ( x) = 1 - 4 x + 3 x ^ {2} - \frac{2 x ^ {3} }{3} + \frac{x ^ {4} }{24} . $$

The Laguerre polynomial $ L _ {n} ^ \alpha ( x) $ is sometimes denoted by $ L _ {n} ( x ; \alpha ) $.

References

[1]	E. Laguerre, "Sur le transformations des fonctions elliptiques" Bull. Soc. Math. France , 6 (1878) pp. 72–78
[2]	V.A. Steklov, Izv. Imp. Akad. Nauk. , 10 (1916) pp. 633–642
[3]	G. Szegö, "Orthogonal polynomials" , Amer. Math. Soc. (1975)
[4]	P.K. Suetin, "Classical orthogonal polynomials" , Moscow (1979) (In Russian)

Comments

Laguerre polynomials can be written as confluent hypergeometric functions (cf. Confluent hypergeometric function) and belong to the classical orthogonal polynomials. They have a close connection with the Heisenberg representation: as matrix elements of irreducible representations and as spherical functions on certain Gel'fand pairs (cf. Gel'fand representation) associated with the Heisenberg group. See the references given in [a1], Chapt. 1, §9.

References

[a1]	G.B. Folland, "Harmonic analysis in phase space" , Princeton Univ. Press (1989)