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
\( \phi(s,a,b,c) = \Omega_{j=1}^\infty e^{a(s+j) + b + cz}\bullet z\\= \lim_{n\to\infty} e^{\displaystyle a(s+1) + b + ce^{\displaystyle a(s+2) + b + ce^{...a(s+n)+b+cz}}} \)
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