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}]

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}]

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

True

Manipulate[Plot[Evaluate@LaguerreL[Sort@n, x], {x, 0, 10}, 
  PlotLegends -> ("LaguerreL[" <> ToString[#] <> ", x]" & /@ Sort[n]), 
  PlotRange -> {-15, 15}], 
{{n, {1}}, Range[10], TogglerBar}]

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}]