Laguerre polynomials: laguerre

laguerre takes as argument an integer n and optionnally 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) = $${\frac{{2n+a-1-x}}{{n}}}$$L(n - 1, a, x) - $${\frac{{n+a-1}}{{n}}}$$L(n - 2, a, x)

These polynomials are orthogonal for the scalar product

< f, g > = $$\int_{{0}}^{{+\infty}}$$f (x)g(x)x^ae^-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