A related discussion on interpolation: factorial and gamma-function

[Gottfried]

For the ongoing discussion about the best way to define a continuous version for the tetration it might be useful to have a look at the following article, which deals with the equivalent problem concerning the gamma-function. See Luschny's Factorial


Although the gamma-function is now well established, its definition as a continuous version of the factorial function has a history of controversity about what might be the most useful and most natural property to obtain by a certain type of interpolation. I'm unable to comment on this properly, but possibly the more experienced participiants of this forum can take some thing inspiring from this. Anyway - the glance, which I get from this text, makes me cautions with assumtions, that "my" preferred interpolation of tetration is the "most natural" or "only useful" one... ;-)

Gottfried

[bo198214]

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

[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

[bo198214]

However as I now see the Hadamard and Luschny definitions hava a major drawback, they dont satisfy
\( (x+1)!=(x+1)x! \)
they merely interpolate n!. So there is no (mentioned) alternative definition of the gamma function that satisfies the above equation.

For tetration we too demand that
\( b[4](x+1)=b^{b[4]x} \)
and that it is not just an interpolation for b[4]n.
However here there seem to be several competing definitions.

[Gottfried]

However as I now see the Hadamard and Luschny definitions hava a major drawback, they dont satisfy
\( (x+1)!=(x+1)x! \)
they merely interpolate n!. So there is no (mentioned) alternative definition of the gamma function that satisfies the above equation.

For tetration we too demand that
\( b[4](x+1)=b^{b[4]x} \)
and that it is not just an interpolation for b[4]n.
However here there seem to be several competing definitions.

upps? I thought(and was sure by reading the article) that for the positive integers there is identity with the "x! = x*(x-1)!" ? Did I miss something?
--- Ah , well, you mean for fractional x? Well - I'll consult the article again.

Thanks for the hint...

Gottfried

[bo198214]

--- Ah , well, you mean for fractional x? Well - I'll consult the article again.

Ya, ya, I meant \( x \) for real numbers and \( n \) for natural numbers.

[Gottfried]

--- Ah , well, you mean for fractional x? Well - I'll consult the article again.

Ya, ya, I meant \( x \) for real numbers and \( n \) for natural numbers.

Just checked. You're right. So the Hadamard/Luschny-function ... not very well configured for the idea of a factorial.

[Catullus]

The article says "If n = 0,-1,-2,... then Gamma(n) becomes infinite.". Could you please elaborate on that. Isn't gamma of zero or a negative integer undefined?

I know in some settings gamma of zero or a negative integer is defined as complex infinity, but if complex infinity equals one divided by zero, then complex infinity times zero would equal one. If complex infinity times zero equals one, then multiplying both sides by two that would mean that complex infinity times zero times two equals two. Zero times two is zero, so that would mean that complex infinity times zero times two would equal complex infinity times zero, which would equal one. So that would make one one equal two.

[JmsNxn]

No, Catullus

The \(\Gamma\) function is a meromorphic function. This means it sends \(\mathbb{C} \to \mathbb{C} \cup \infty\). Infinity is perfectly fine in this instance, because we are referring to it on the Riemann sphere. All is good. Look up the notion of a pole, the Gamma function has simple poles at the negative integers.

[Gottfried]

It says "If n = 0,-1,-2,... then Gamma(n) becomes infinite.". Gamma of zero or a negative integer is undefined, not infinite. For example, . One divided by zero is not infinite, it is undefined.

Peter Luschny has once explained this to me with the (standard) concept of limiting towards an infinitesimal interval. See the approximation of the quotient in \( \lim_{h \to 0} { \Gamma (0 \pm h) \over \zeta(1 \pm h) } \) :

Code:

   .
h=0.001;[gamma(0+h)/zeta(1+h),gamma(0-h)/zeta(1-h)]
%9 = [0.998847149536, 1.00115601566]

h=1e-5;[gamma(0+h)/zeta(1+h),gamma(0-h)/zeta(1-h)]
%11 = [0.999988455845, 1.00001154447]

h=1e-6;[gamma(0+h)/zeta(1+h),gamma(0-h)/zeta(1-h)]
%13 = [0.999998845570, 1.00000115443]

h=1e-12;[gamma(0+h)/zeta(1+h),gamma(0-h)/zeta(1-h)]
%15 = [0.999999999999, 1.00000000000]

...