Laguerre Polynomials
Laguerre Polynomials
Table[LaguerreL[k,x],{k,0,6}]
Plot[Evaluate[Table[LaguerreL[k,x],{k,0,6}]],{x,0,6}];
Laguerre's equation is
xy''+(1-x)y'+Ly=0 0<x<Infinity
If we multiply through by Exp[-x] we get
[ xExp[-x]y' ]' +LExp[-x] y=0 (Check this!)
Here r(x)=xExp[-x], q(x)=0, p(x)=Exp[-x]
This is a singular S-L problem with endpoint conditions y(0+) is bounded and
Exp[-x]y(x)->0 as x->Infinity.
You may also see
(xy')' + [ L+(2-x)/4 ]y=0 0<x<Infinity
with endpoint condition y is bounded as x->Infinity and as x->0.
Here the eigenfunctions are the Laguerre polynomials times Exp[-x/2].
The eigenvalues of Laguerre's equation are the nonnegative integers n. The eigenfunctions are the Laguerre polynomials.
Note that the eigenvalues are real and satisfy L->Infinity.
These polynomials DO solve the differential equation. Check one of them.
Simplify[x D[LaguerreL[2,x],{x,2}]+(1-x)D[LaguerreL[2,x],x]+2LaguerreL[2,x]]Check the first seven:
Table[x D[LaguerreL[2,x],{x,2}]+(1-x)D[LaguerreL[2,x],x]+2LaguerreL[2,x],{k,0,6}]
Simplify[%]
By one of the major Sturm-Liouville theorems, the Laguerre polynomials are orthogonal with respect to the weight Exp[-x]. Check one.
Integrate[Exp[-x]LaguerreL[1,x]LaguerreL[2,x],{x,0,Infinity}]Check many at a time:
Table[Integrate[Exp[-x]LaguerreL[k,x]LaguerreL[m,x],{x,0,Infinity}],{k,0,3},{m,0,3}];
MatrixForm[%]The Laguerre polynomials have many interesting properties. Using what is above you can't see one that will prove helpful, so I'll show you....
Integral[Exp[-x](Laguerre[k,x])^2]=(n!)^2
The Laguerre polynomials, like the Legendre and Hermite polynomials, have a relationship involving repeated derivatives.
LaguerreL[k,x]=Exp[x]/n! D[x^n Exp[-x], {x,k}]
Does it seem to work?
Table[Exp[x]/k! D[x^k Exp[-x], {x,k}],{k,0,4}]
Simplify[%]
Table[LaguerreL[k,x],{k,0,4}]