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)