[2014] Beyond Gamma and Barnes-G

[tommy1729]

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

[MphLee]

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 \( \neq 1 \)

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