A related discussion on interpolation: factorial and gamma-function

[Gottfried]

Wow, thats really informative. I never before heard about the Hadamard interpolation for the factorials.

Hmm, yes, when this discussion started in de.sci.mathematik last year, I was quite impressed. On the other hand: there was no evidence, that this new (and old) definitions are of a certain use. It is a pity, that until now no such evidence could be shown. For instance: conversion of current formulae containing the gamma-function into such containing the other definition and the example, that this is a superior formula (more natural, more smooth,...)

But, well, I'd still be interested to see such things, and I hope, Peter Luschny will work on this further.
It is a permanent experience to me, that generalizations of known formulae can exhibit important basic properties of a mathematicla relation or of fundamental principles. In this view I like for instance the formulae for my tetra-geometric series as in my earlier posts: this generalization embeds a simple property of the geometric series, which seems to be not even worth to be mentioned (since it is so tiny), into a rule of a general behaviour of series for each height of tetration (at least positive integer height), and I'd say, in this regard it has the potential to be one of the "classical" properties in the field of series.
One of the criteria, which type of interpolation for tetration will be the "most natural" will surely be, which type provides the most interesting and generalizable properties in the usual context of powerseries.

Gottfried