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 $(f(f(x))=xf(x))$ on ${\mathbb{R}}^{+}$, leading for continuous $f$ to $f$ = $\mathrm{\varnothing }$ or the power function $f(x)$ = $x^{\alpha }$, $\alpha =\sqrt{5}+1)\mathrm{/}2$. 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.