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[2014] Beyond Gamma and Barnes-G
#1
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
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#2
Mhh maybe I have made some errors but the solution of the eqation seems to have some problems in the first values... that should make it not determinate for natural values


Let assume that then

because of the limit

is not determinate (multivalued?)

I'm not sure how to continue in this case... maybe we can start by fixing the value of and continue with the recursion






...




So actually the solution is of the form for some


?
------

Btw I see the relation with the Bennet's commutative Hyperoperations (denote it by )!



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

--------------

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

Given or


MSE MphLee
Mother Law \((\sigma+1)0=\sigma (\sigma+1)\)
S Law \(\bigcirc_f^{\lambda}\square_f^{\lambda^+}(g)=\square_g^{\lambda}\bigcirc_g^{\lambda^+}(f)\)
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