The ultimate sanity check

[Gottfried]

(...)
Trying to read Gottfried... doesn't the problem lies into going from finite approximations to infinite "divergent summation techniques"? Maybe is there where we lack of formal proofs that those semi-group identities holds?

 Hi MphLee -

 the problem of finite matrices / power series, where infinite are required is really a point.  
There are two aspects:

 1) can we approximate from truncated series? For convergent series this is common practice; for specific types of divergence, for instance alternating geometric serie, this is also common. Especially if we know the "form" of the "general term" (as L. Euler coined it) in the case the matrix/the powerseries would be infinite.  
But with our exponential-function it has been proven that fractional iterates produce so badly diverging series that there is much area to explore here... Well, for instance for the much divergent alternatig series of factorials Euler found a closed form which allows to handle that sometimes.         
But the coefficients in the fractional powers of the Carlemanmatrix for \( e^x-1 \) diverge even more... so any summation method which extrapolates a limit from a finite number of terms must come with a very diplomatic exposition of the conjectured values... ;-)    But if inserting such an approximated value in the series again, when then a reasonable value comes out, then we have at least an argument that our approximated value is not completely off-the-road... and may be published (with all cautions)

2) The second aspect it much more difficult, and I have very rarely a handling for this: if the Carlemanmatrix is not triangular but square, then it might absolutely crap to try to extrapolate from finite sizes to the unknown infinite size case. It seems, that for instance Andrew's ansatz and his extrapolation to matrices of infinite size is systematically wrong by some tiny difference (this problem has also Peter Walker in his article considered). (I can say more later, I just have to interrupt this).


Gottfried