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LaguerreL - Laguerre function

Calling Sequence

LaguerreL(n, a, x)

Parameters

n

-

algebraic expression

a

-

(optional) nonrational algebraic expression or rational number

x

-

algebraic expression

Description

• 

The LaguerreL function computes the nth Laguerre polynomial.

• 

If the first parameter is a non-negative integer, the LaguerreL function computes the nth generalized Laguerre polynomial with parameter a evaluated at x.

  

If a is not specified, LaguerreL(n, x) computes the nth Laguerre polynomial which is equal to LaguerreL(n, 0, x).

• 

The generalized Laguerre polynomials are orthogonal on the interval 0, with respect to the weight function wx=ⅇxxa. They satisfy:

0wtLaguerreLm,a,tLaguerreLn,a,tⅆt={0nmΓn+a+1n!n=m

• 

For positive integer a, the relationship for LaguerreL(n, a, x) and LaguerreL(n, x) is the following.

LaguerreLn,a,x=1aaxaLaguerreLn+a,x

  

Some references define the generalized Laguerre polynomials differently than Maple. Denote the alternate function as altLaguerreL(n, a, x). It is defined as follows:

altLaguerreLn,a,x=axaaltLaguerreLn,x

altLaguerreLn,x=n!LaguerreLn,x

  

For general positive integer a, the relationship for Maple's LaguerreL and altLaguerreL is the following.

altLaguerreLn,a,x=1an!LaguerreLna,a,x

• 

Laguerre polynomials satisfy the following recurrence relation:

LaguerreL0,a,x=1,

LaguerreL1,a,x=x+1+a,

LaguerreLn,a,x=2n+a1xnLaguerreLn1,a,xn+a1nLaguerreLn2,a,x,forn>1.

• 

For n not equal to a non-negative integer, the analytic extension of the Laguerre polynomial is given by:

LaguerreLn,a,x=binomialn+a,nKummerMn,a+1,x

Examples

LaguerreL3,x

LaguerreL3,x

(1)

simplify,'LaguerreL'

13x+32x216x3

(2)

LaguerreL3,12,x

LaguerreL3,12,x

(3)

simplify,'LaguerreL'

516158x+54x216x3

(4)

LaguerreL3.1,1.2

0.7174310784

(5)

LaguerreL2.1,1.2,3.4

1.498106063

(6)

Using the alternate definition for the Laguerre polynomials:

altLaguerreL:=n,a,x→1an!LaguerreLna,a,x:

altLaguerreL3,1,x

6LaguerreL2,1,x

(7)

simplify,'LaguerreL'

3x2+18x18

(8)

See Also

ChebyshevT, ChebyshevU, GAMMA, GegenbauerC, HermiteH, JacobiP, LegendreP, orthopoly[L], simplify


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