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}]
