Background/context: Having had some numerical success trying to calculate a family of fractional logarithms.
What would be the most reasonable basis of functions to try and express a fractional logarithm in ?
An ordinary family of polynomial seems unsuitable to build linear combinations with : the function is far too "close-to-linear" or what we shall call it.
My initial guess would be a linear function together with a set of fastly decaying functions, for example: $$\{x,1,x^{-1},\cdots ,x^{-k}\}$$ Maybe we can tie it to a partial Laurent expansion and some properties in Complex Analysis? Or will we need to dig deeper?