Hi,
in a thread in math.stackexchange.com (limit of a recursive function) I came over the question of what could be a closed form for
\(
z_{k+1} = z_k - \frac 1{z_k}\\
z_0 = z \in \mathbb{R}\\
t = f(z_0) = \lim_{n \rightarrow \infty} z_n\\
\)
I fiddled a bit with it and its reverse operation, finding interesting properties, for instance the existence of periodic points of any order, after another contributor showed that the only 1-periodic point (=fixpoint) would be infinity. (see https://math.stackexchange.com/a/4056192/1714)
To extend my knowledge about this sequence/function beyond that MSE-discussion I pondered the possibility of fractional iteration (or as one might say: indefinite summation) but couldn't find a promising ansatz to establish such a routine. However, further thinking showed, that the recursive expression could as well be as iteration of the function g(z)= 2*sinh (log(z)) , and since we had discussions here about iteration of 2*sinh(z) there might as well be an idea for the fractional iteration of the form g(z).
Someone out here with an idea? (Feel free to contribute to the thread in MSE)
Gottfried
in a thread in math.stackexchange.com (limit of a recursive function) I came over the question of what could be a closed form for
\(
z_{k+1} = z_k - \frac 1{z_k}\\
z_0 = z \in \mathbb{R}\\
t = f(z_0) = \lim_{n \rightarrow \infty} z_n\\
\)
I fiddled a bit with it and its reverse operation, finding interesting properties, for instance the existence of periodic points of any order, after another contributor showed that the only 1-periodic point (=fixpoint) would be infinity. (see https://math.stackexchange.com/a/4056192/1714)
To extend my knowledge about this sequence/function beyond that MSE-discussion I pondered the possibility of fractional iteration (or as one might say: indefinite summation) but couldn't find a promising ansatz to establish such a routine. However, further thinking showed, that the recursive expression could as well be as iteration of the function g(z)= 2*sinh (log(z)) , and since we had discussions here about iteration of 2*sinh(z) there might as well be an idea for the fractional iteration of the form g(z).
Someone out here with an idea? (Feel free to contribute to the thread in MSE)
Gottfried
Gottfried Helms, Kassel