Matrix Operator Method

[bo198214]

Just derived a method to compute exact entries for powers of the (square) matrix-operator for T-tetration.
It is applicable to positive integer powers only, but for any base.
...

Code:

T^2 = U*dV(b^^0) * T*dV(b^^1)
T^3 = U*dV(b^^0) * U*dV(b^^1) * T*dV(b^^2)
...
T^h = prod_{k=0}^{h-2} (U * dV(b^^k))
           * (T * dV(b^^(h-1)))

So the finding is that though the matrix multiplication of the T's is infinite, the result is expressible in finite terms (polynomials) of \( \text{log}(b) \)?

This finding is interesting, because in my matrix-method I had to use fixpoint-shift to get exact entries even for integer powers, and since the fixpoints for T-tetration are real only for a small range of bases we had to deal with complex-valued U-matrices when considering the general case. Here we do not need a fixpoint-shift.

I did not check how this computation is related to Ioannis Galidakis' method for exact entries yet, but I think, this is interesting too.

That would be useful to compare. Can you derive a recurrence from your matrix formula?