Comparing the derivations for formulae for fractional iteration using derivatives vs. fractional powers of matrices, I came to a consideration, which took me last days, but is still vague for me.

I recall, that the eigensystem-decomposition provides a matrix of coefficients, which allow to write the general iteration of height h and "start-parameter" x, say , and the h'th iterate as function of two parameters (using u=log(t)) and are different (powerseries-defined) functions of x, also dependent on u:

for instance u=2

where, if we derive wrt x we use derivatives according to derivation of powerseries; but if we derive wrt h, we have to use derivation of dirichlet-like series, which is quite different.

Well, if I fix x here, denote and write the Euler-Maclaurin-formula for the sum of derivatives expand and sum, I get formally the correct result for F(1).

So basically I don't have any problem here (besides that I'll have to become more familiar with this), but since I didn't see this difference mentioned anywhere, I wonder, whether I've it simply overlooked or whether it is even trivial...

[update] Well, it's not that I didn't understand derivatives, it's only, that I was extremely focused on my matrix-concept, and now get aware of some more very simple relations. So maybe it's only a "whoops"-reminder... [/update]

Scratching head...

Gottfried

I recall, that the eigensystem-decomposition provides a matrix of coefficients, which allow to write the general iteration of height h and "start-parameter" x, say , and the h'th iterate as function of two parameters (using u=log(t)) and are different (powerseries-defined) functions of x, also dependent on u:

for instance u=2

where, if we derive wrt x we use derivatives according to derivation of powerseries; but if we derive wrt h, we have to use derivation of dirichlet-like series, which is quite different.

Well, if I fix x here, denote and write the Euler-Maclaurin-formula for the sum of derivatives expand and sum, I get formally the correct result for F(1).

So basically I don't have any problem here (besides that I'll have to become more familiar with this), but since I didn't see this difference mentioned anywhere, I wonder, whether I've it simply overlooked or whether it is even trivial...

[update] Well, it's not that I didn't understand derivatives, it's only, that I was extremely focused on my matrix-concept, and now get aware of some more very simple relations. So maybe it's only a "whoops"-reminder... [/update]

Scratching head...

Gottfried

Gottfried Helms, Kassel