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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.

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}]

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.}}

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