How to check whether Laguerre polynomials are orthogonal?

Question

I've got the problem with checking if Laguerre polynomials for n=1,...,10 are orthogonal.

I have to create the list of these polynomials, then create the matrix of integrals from 0 to infinity. Something like:

M=Integrate[LaguerreL[i,x] LaguerreL[j,x] Exp[-x], {x,0,Infinity}] 

And in the end I have to draw the dynamic drawing of these polynomials so that if I choose on graph n, from 0 to 20, the correct polynomial will be drawn with its derivative.

Answer 1

Integrate[LaguerreL[i, x] LaguerreL[j, x] Exp[-x], {x, 0, Infinity}, 
 Assumptions -> Element[{i, j}, Integers] && j > i > 0]

0

n = 10;
Outer[Integrate[LaguerreL[#, x] LaguerreL[#2, x] Exp[-x], {x, 0, ∞}] &, 
  Range[n], Range[n]] == IdentityMatrix[n]

True

Manipulate[Show[Plot[Evaluate@LaguerreL[Sort@n, x], {x, 0, 10}, 
   PlotLegends -> ("LaguerreL[" <> ToString[#] <> ", x]" & /@ Sort[n]), 
   PlotRange -> {-15, 15}], 
  Plot[Evaluate[D[LaguerreL[Sort@n, z], z] /. z -> x], {x, 0, 10}, 
   PlotLegends -> ("D[LaguerreL[" <> ToString[#] <> ", x], x]" & /@ Sort[n]), 
   PlotStyle -> Dashed]],
 {{n, {5, 10, 17}}, Range[0,20], TogglerBar}]

Answer 2

Table[
 NIntegrate[LaguerreL[i, x] LaguerreL[j, x] Exp[-x], {x, 0, Infinity}],
 {i, 10},
 {j, 10}
] // Chop // Quiet
MatrixForm@%
Manipulate[
 Plot[
  {#, D[#, x]} &@LaguerreL[n, x] // Evaluate,
  {x, 0, 10},
  Frame -> True,
  BaseStyle -> {11, FontFamily -> Times},
  PlotLabel -> StringForm["n=``", n]
 ],
{n, 0, 20, 1, PopupMenu}
]

{{1., 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 1., 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 1., 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 1., 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 1., 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 1., 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 1., 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 1., 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 1., 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 1.}}

$\left( \begin{array}{cccccccccc} 1. & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1. & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1. & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1. & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1. & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1. & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1. & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1. & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1. & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1. \\ \end{array} \right)$