Discrete convolution power

In my previous question we have discussed the posibility of various definition of convolution of power function within Mathematica. Now the question is "How to define convolution power in Mathematica ?" We will mainly be targeted on power function in our examples as well... The $$n$$-th convolution power is $$n$$-th convolution with itself. So, if we have convolution power $$n=2$$ then we have to get just convolution of convolution and so on.

The problem is: Write the program which allows take the $$n>1$$-th convlotion power of power function just changing the variable $$n$$ within the program.

Here I attach my code for common convolution of power function $$s^m$$ defined for $$s\geq 0$$ and zero otherwise.

To solve the problem we are free just to revise the following code

f[m_, s_] := Piecewise[{{s^m, s >= 0}, {0, True}}];

F[n_, m_] := Sum[f[m, n - k]*f[m, k], {k, -Infinity, +Infinity}];

T[n_, k_] := F[n - k, k];

Column[Table[T[n, k], {n, 0, 12}, {k, 0, n}], Left]


Such that it will be introduced some varaiable that will mean the convolution power. For convolution power read more at Wikipedia.

• Looks like a job for Fold. – Henrik Schumacher Mar 25 at 7:15