I am interested in q-differential equations of the form

$p(f(z), f(qz),\dots,f(q^kz))=0$

where $p$ is a polynomial and $k$ an nonnegative integer. I wonder about the closure properties of the class of (formal) powers series satisfying such an equation. A bit is known when $p$ is required to be linear, see Section 3 of "A Mathematica package for q-holonomic sequences and power series" by Manuel Kauers and Christoph Koutschan.

In particular, if $f$ and $g$ satisfy a $q$-ADE, do $f+g$, $f\cdot g$ and $f\circ h$ for suitably simple $h$, too? And if so, what is the $k$ in the resulting equations?