Consider the functions defined by the equations and conditions
$$Re(z) > 2$$
$$f(i+1) > f(i)-2$$
$$ f(z) = \sum_{1<i}^{\infty} f(i)^{\dfrac{z-i+1}{i}}$$
Does that make sense ? Does that converge ?
Suppose it does not make sense for some divergent solutions, does there exist a kind of continuation ?
What solutions exist ? How many solutions exist ? What are the asymptotics ?
Is it always analytic for all $2 < Re(z)$ ?
In particular for $f(i)$ being real and positive.