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orthopoly

  

L

  

Laguerre polynomial

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

L(n, a, x)

L(n, x)

Parameters

n

-

non-negative integer

a

-

rational number greater than -1 or nonrational algebraic expression

x

-

algebraic expression

Description

• 

The L(n, a, x) function computes the nth generalized Laguerre polynomial with parameter a evaluated at x.

  

In the two argument case, L(n, x) computes the nth Laguerre polynomial which is equal to L(n, 0, x).

• 

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

0wtLm,a,tLn,a,tⅆt=0nmΓa+n+1n!n=m

• 

For positive integer a, Ln,a,x is related to Ln,x by:

Ln,a,x=1aⅆaⅆxaLn+a,x

  

Some references define the generalized Laguerre polynomials differently from Maple. Denote the alternate function as altLn,a,x. It is defined as:

altLn,a,x=ⅆaⅆxaaltLn,x

altLn,x=n!Ln,x

  

For a general positive integer a, the Maple orthopoly[L] function is related to altL by:

altLn,a,x=1an!Lna,a,x

• 

Laguerre polynomials satisfy the following recurrence relation.

L0,a,x=1,

L1,a,x=x+1+a,

Ln,a,x=2n+a1xLn1,a,xnn+a1Ln2,a,xn,for n>1.

Examples

withorthopoly:

L3,x

13x+32x216x3

(1)

L15,5

19982258571307674368

(2)

L2,1,x

33x+12x2

(3)

L11,17,58

40912499266488426014273327119645385619965562847232

(4)

Using the alternate definition for the Laguerre polynomials:

altLn,a,x→1an!orthopoly[L]na,a,x:

altL3,1,x

3x2+18x18

(5)

See Also

GAMMA

LaguerreL

 


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