Laguerre polynomials: laguerre

### 5.31.3 Laguerre polynomials: laguerre

laguerre takes as argument an integer n and optionally a variable name (by default x) and a parameter name (by default a).
laguerre returns the Laguerre polynomial of degree n and of parameter a.
If L(n,a,x) denotes the Laguerre polynomial of degree n and parameter a, the following recurrence relation holds:

L(0,a,x)=1, L(1,a,x)=1+a−x, 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)xae−xdx

Input :

laguerre(2)

Output :

(a`^`2+-2*a*x+3*a+x`^`2+-4*x+2)/2

Input :

laguerre(2,y)

Output :

(a`^`2+-2*a*y+3*a+y`^`2+-4*y+2)/2

Input :

laguerre(2,y,b)

Output :

(b`^`2+-2*b*y+3*b+y`^`2+-4*y+2)/2