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Algoritmization of parametric derivation - Xorter - 04/12/2017

I have a really strange f function (that I would like to take into Carleman matrices) so I need the derivative of its powers. We have just know about the function that:


where a is a constant.
I am interested in the following expression:

A recursive formula is not the best, but really enough. Not to be in PARI/gp is not the best, but enough.
(It is important for me to invastigate the infinite Taylor series with argument infinity.)