Explicit expression for the generating function F(x) for general s≥1

Derive an explicit closed-form expression for the generating function F(x) = Σ_{n=1}^{∞} f_n^2 x^n associated with the Z-Hamiltonian systems, where f_n = √A_n^{(s)} and A_n^{(s)} are the Fuss–Catalan numbers, valid for all s ≥ 1, extending beyond the cases s=1 and s=2 where explicit forms are available.

Background

The invariant manifold reduction in the Z-Hamiltonian analysis relies on the generating function F(x) built from Fuss–Catalan numbers. For s=1 and s=2, explicit expressions are known and have been used in prior works. For general s≥1, the paper works with a differential equation for F and the relation x(F) rather than a closed form.

Obtaining an explicit expression for F(x) across all s≥1 would enhance analytic control and potentially broaden the class of exact solutions tractable within this framework.