# closure properties of q-differential equations

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?

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Martin, these questions (maybe, from a less algorithmic point of view) are addressed by Lucia di Vizzio. –  Wadim Zudilin Jun 19 '11 at 23:37
Wadim, could you give me a slightly more precise pointer? Lucia's articles are not particularly easy reading for me... –  Martin Rubey Jun 24 '11 at 15:47
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