Generalized phi(s,a,b,c)

[tommy1729]

Hello everyone.

James phi function reminded me of earlier ideas I had for solving tetration.
I got stuck with those ideas but phi has given me new courage.

James his phi function is part of the generalization that I considered many years ago.



Notice the parameters a,b,c are very closely related !

In fact these functions seem very related.

is clearly tetration.

i tried to take limits towards zero for the parameters to arrive at tetration.

Also the derivatives act similar like those of tetration. Hence ideas of continu sum and products arose. See also many threads including mike3 and others.

Is James Nixon's phi and as said the derivative is very much like that of tetration.

James his phi had a closed form.

Does this generalization - apart from tetration itself perhaps -  have other closed forms ? And are those entire or analytic ?

regards

tommy1729

[MphLee]

Also the derivatives act similar like those of tetration. Hence ideas of continu sum and products arose. See also many threads including mike3 and others.

Hi, can you point me to some of those threads ure referring to?

[tommy1729]

Also the derivatives act similar like those of tetration. Hence ideas of continu sum and products arose. See also many threads including mike3 and others.

Hi, can you point me to some of those threads ure referring to?

Here are some of them :

https://math.eretrandre.org/tetrationfor...p?tid=1028

https://math.eretrandre.org/tetrationfor...p?tid=1005

https://math.eretrandre.org/tetrationfor...hp?tid=964

https://math.eretrandre.org/tetrationfor...hp?tid=880

https://math.eretrandre.org/tetrationfor...hp?tid=814

https://math.eretrandre.org/tetrationfor...hp?tid=768

https://math.eretrandre.org/tetrationfor...hp?tid=661

https://math.eretrandre.org/tetrationfor...hp?tid=582

https://math.eretrandre.org/tetrationfor...hp?tid=535

https://math.eretrandre.org/tetrationfor...hp?tid=500

https://math.eretrandre.org/tetrationfor...hp?tid=459

https://math.eretrandre.org/tetrationfor...hp?tid=732

https://math.eretrandre.org/tetrationfor...hp?tid=523

https://math.eretrandre.org/tetrationfor...hp?tid=516

https://math.eretrandre.org/tetrationfor...hp?tid=437

https://math.eretrandre.org/tetrationfor...hp?tid=404

https://math.eretrandre.org/tetrationfor...hp?tid=370

That should give you an idea how continu sum is related.

regards

tommy1729

[JmsNxn]

Here's the closed form tommy,



This converges for --and is holomorphic on these domains; and converges to the same function for all .

[tommy1729]

by analogue,



This converges for --and is holomorphic on these domains; and converges to the same function for all .

For instance this solves f(s+1) = - s + exp(f(s)) by letting a = -1.
( this too would create a NBLR type solution to tetration but with similar problems I think )

Analytic continuations are perhaps not possible for your case (or my analogue) in attempt to go from Re(a) < 0 to Re(a) > 0 or vice versa.

In fact there is a huge gap in my understanding about continuations for infinite compositions. Or Riemann surfaces of infinite compositions.
But I think a natural boundary occurs for Re(a) = 0 in both our cases.

Nevertheless I am inspired by this.

regards

tommy1729

[tommy1729]

To clarify 

let 
f(s+1) = exp(f(s)) + a*s + b
g(s+1) = exp(g(s)) - a*s - b

then 

f(s+1) + g(s+1) = exp(f(s)) + exp(g(s))

Now assume g(s+1) = g(s) ( g is then no longer entire but may be analytic )

For some a and b and f and g this might be interesting.
Or use infinitesimals.

I know not very formal, just sketchy ideas.

Another crazy idea is the generalization with similar functions a,b,c :

a(s+1) + b(s+1) + c(s+1) = exp(a(s+1)) + exp(b(s+1)) + exp(c(s+1))
a(s+2) + b(s+2) + c(s+2) = exp^[2](a(s+1)) + exp^[2](b(s+1)) + exp^[2](c(s+1))

D(s) = a(s) + b(s) + c(s).

And then somehow get tetration from D(s).

Im talking analytic tetration here ofcourse.

crazy ideas :p

regards

tommy1729

[JmsNxn]

....

Analytic continuations are perhaps not possible for your case (or my analogue) in attempt to go from Re(a) < 0 to Re(a) > 0 or vice versa.

In fact there is a huge gap in my understanding about continuations for infinite compositions. Or Riemann surfaces of infinite compositions.
But I think a natural boundary occurs for Re(a) = 0 in both our cases.

....

tommy1729

Yes, I'd have to agree with you as it being a natural boundary. When we flip to all we get is the equation,



From,