Generalized phi(s,a,b,c) tommy1729 Ultimate Fellow Posts: 1,739 Threads: 382 Joined: Feb 2009 02/04/2021, 01:17 PM (This post was last modified: 02/04/2021, 01:20 PM by 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 $\phi(s,a,b,c)$ that I considered many years ago. $\phi(s+1,a,b,c) = \exp(a s + b + c \phi(s,a,b,c))$ Notice the parameters a,b,c are very closely related ! In fact these functions seem very related. $\phi(s+1,0,0,1) =\exp(0s+0+1\phi(s,0,0,1))$ 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. $\phi(s+1,1,0,1) =\exp(as+b+c \phi(s,1,0,1))$ 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 Long Time Fellow Posts: 367 Threads: 28 Joined: May 2013 02/04/2021, 06:25 PM (02/04/2021, 01:17 PM)tommy1729 Wrote: 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? MSE MphLee Mother Law $$(\sigma+1)0=\sigma (\sigma+1)$$ S Law $$\bigcirc_f^{\lambda}\square_f^{\lambda^+}(g)=\square_g^{\lambda}\bigcirc_g^{\lambda^+}(f)$$ tommy1729 Ultimate Fellow Posts: 1,739 Threads: 382 Joined: Feb 2009 02/05/2021, 12:59 AM (02/04/2021, 06:25 PM)MphLee Wrote: (02/04/2021, 01:17 PM)tommy1729 Wrote: 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 Ultimate Fellow Posts: 1,056 Threads: 121 Joined: Dec 2010 02/06/2021, 12:18 AM (This post was last modified: 02/06/2021, 03:53 AM by JmsNxn.) Here's the closed form tommy, $ \phi(s,a,b,c) = \Omega_{j=1}^\infty e^{a(s-j) + b + cz}\bullet z\\ = \lim_{n\to\infty} e^{\displaystyle a(s-1) + b + ce^{\displaystyle a(s-2) + b + ce^{...a(s-n)+b+cz}}}$ This converges for $\Re(a) > 0, s,b,c \in \mathbb{C}$--and is holomorphic on these domains; and converges to the same function for all $z\in\mathbb{C}$. tommy1729 Ultimate Fellow Posts: 1,739 Threads: 382 Joined: Feb 2009 02/07/2021, 05:03 PM (This post was last modified: 02/07/2021, 05:04 PM by tommy1729.) by analogue, $\phi(s,a,b,c) = \Omega_{j=1}^\infty e^{a(s+j) + b + cz}\bullet z\\= \lim_{n\to\infty} e^{\displaystyle a(s+1) + b + ce^{\displaystyle a(s+2) + b + ce^{...a(s+n)+b+cz}}}$ This converges for $\Re(a)<0, s,b,c \in \mathbb{C}$--and is holomorphic on these domains; and converges to the same function for all $z\in\mathbb{C}$. 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 Ultimate Fellow Posts: 1,739 Threads: 382 Joined: Feb 2009 02/07/2021, 05:37 PM (This post was last modified: 02/07/2021, 06:02 PM by 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 Ultimate Fellow Posts: 1,056 Threads: 121 Joined: Dec 2010 02/08/2021, 12:30 AM (This post was last modified: 02/08/2021, 12:32 AM by JmsNxn.) (02/07/2021, 05:03 PM)tommy1729 Wrote: .... 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 $\Re(a) < 0$ all we get is the equation, $ \psi(s-1,a,b,c) = e^{as + b + c \psi(s,a,b,c)}$ From, $ \psi(s,a,b,c) = \Omega_{j=1}^\infty e^{a(s+j) + b + cz}\bullet z = \phi(-s,-a,b,c)\\$ « Next Oldest | Next Newest »

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