Hirsch meets Fibonacci and Narayana type variants

For functions $f$ of a continuous variable in $\mathbb{R}^{+}$ we show that the Hirsch function $h_f$ equals $f$ iff $(f(f(x)) = x f(x))$ on $\mathbb{R}^{+}$, leading for continuous $f$ to $f$ = $\emptyset$ or the power function $f(x)$ = $x^{\alpha}$, $\alpha= \sqrt{5} +1)/2$. For functions of a discrete positive variable in $\mathbb{R}^{+}$, we show that $h_f$ = $f$ implies that only the trivial function $f$ = {(1,1)} satisfies this. We also study the problem $h_f = f \circ f$ and for $f = g \circ g, h_f = g$ leading to the zero function or another power law in the continuous variable case and again to $f$ = {(1,1)} in the discrete variable case. Both problems involve the study of variants of the Fibonacci sequence for which non-trivial identities are proved and applied in the solution of the above problems.