myRainbow = Function[x, Blend[{Purple, Blue, Green, Yellow, Red}, x]];

nd[y_, a_, b_, n_] :=
  (Binomial[b - a + 1, y] *
     Sum[((-1)^i)*Binomial[y, i]*((y - i)/(b - a + 1))^n, {i, 0, y}]);
ArrayPlot[Table[nd[3, 1, K, n], {K, 1, 15}, {n, 1, 15}],
  ColorFunction -> myRainbow, DataReversed -> True]

DiscretePlot3D[nd[3, 1, k, n], {n, 1, 15}, {k, 1, 15},
  PlotRange -> {0, 1}, ExtentSize -> Full, ColorFunction -> myRainbow]

rainbow[z_] := Blend[{Purple, Blue, Green, Yellow, Red}, z]
rainbow[_, _, z_] := rainbow[z]

DiscretePlot3D[
 nd[3, 1, k, n], {n, 1, 15}, {k, 1, 15},
 PlotRange -> {0, 1},
 ExtentSize -> Full,
 ColorFunction -> rainbow
 ]

GeneralUtilities`PrintDefinitions[DiscretePlot3D]]

ColorData[cf, #3] &

cf

Plot3D[x^2 + y^2, {x, -1, 1}, {y, -1, 1}, ColorFunction -> "Rainbow"]

Plot3D[x^2 + y^2, {x, -1, 1}, {y, -1, 1},
 ColorFunction -> ColorData["Rainbow"]]

Plot3D[x^2 + y^2, {x, -1, 1}, {y, -1, 1},
 ColorFunction -> (ColorData["Rainbow"][#] &)]

# &[1, 2, 3]
(* 1 *)

Plot3D[x^2 + y^2, {x, -1, 1}, {y, -1, 1},
 ColorFunction -> (ColorData["Rainbow"][#3] &)]

ColorFunction -> (myRainbow[#3]&)