# Tetration Forum

Full Version: [2014] Beyond Gamma and Barnes-G
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Most of you here are familiar with the Gamma function , Barnes-G function , the K-function , the double gamma function etc.

But in the spirit of my generalized distributive law / commutative assosiative hyperoperators I was wondering about

f(z+1) = f(z)^ln(z)

It makes sense afterall :

f1(z+1) = z + f1(z) leads to the triangular numbers.

f2(z+1) = z f2(z) leads to the gamma function.

f3(z+1) = f3(z)^ln(z)

Notice f_n(z) = exp^[n]( ln^[n]f_n(z) + ln^[n](z) )

So is there an integral representation for f3 ?

How does it look like ?

Analogues of Bohr-Mullerup etc ?

Hence the connection to the hyperoperators.

regards

tommy1729
Mhh maybe I have made some errors but the solution of the eqation $f(x+1)=f(x)^{ln(x)}$ seems to have some problems in the first values... that should make it not determinate for natural values

Let assume that $f(0)=\beta\gt 1$ then

$f(1)=\beta^{ln(0)}=\beta^{-\infty}=0$ because of the limit

$f(2)=0^{ln(1)}=0^0$ is not determinate (multivalued?)

I'm not sure how to continue in this case... maybe we can start by fixing the value of $f(2)=\gamma$ and continue with the recursion

$f(3)=\gamma^{ln(2)}$

$f(4)=\gamma^{ln(2)ln(3)}$

...

$f(2+n+1)=\gamma^{{\Pi}_{i=2}^{2+n} ln(i)}$

So actually the solution is of the form for some $\gamma$

$f(z+1)=\gamma^{{\Pi}_{i=2}^{z} ln(i)}$?
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Btw I see the relation with the Bennet's commutative Hyperoperations (denote it by $a\odot^e_i b=a\odot_i b$)!

$f_i(z+1)=f_i(z)\odot_i z$

This is like a kind of "Hyper-factorial" that uses the Bennet's Hyperops..

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Edit: I think that the correct forumla is something of the form

Given $f(z_0)=\alpha \neq 0$ or $\neq 1$

$f(z_0+n+1)=\exp_\alpha ({\Pi}_{i=0}^{n} ln(z_0+i))=\alpha^{{\Pi}_{i=0}^{n} ln(z_0+i)}$