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11.5.3 Laguerre polynomials

The Laguerre polynomial of degree n and parameter a satisfy the following recurrence relation:

     
  L(0,a,x)=1         
L(1,a,x)=1+ax         
L(n,a,x)
=
2n+a−1−x
n
L(n−1,a,x)−
n+a−1
n
L(n−2,a,x)
         

These polynomials are orthogonal for the scalar product

  ⟨ f,g⟩=
+∞


0
f(x)g(x)xaexdx.

The laguerre command finds the Laguerre polynomials.

Examples

laguerre(2)
     
1
2
a2ax+
3
2
a+
1
2
x2−2 x+1
          
laguerre(2,y)
     
1
2
a2ay+
3
2
a+
1
2
y2−2 y+1
          
laguerre(2,y,b)
     
1
2
b2by+
3
2
b+
1
2
y2−2 y+1
          

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