Summability of q-hypergeometric terms when q is a root of unity

Investigate and determine necessary and sufficient conditions for oy-summability of oy-hypergeometric terms over F(y) when oy is the q-shift operator with oy(y) = qy and q a root of unity; specifically, establish criteria deciding whether a given oy-hypergeometric term T equals Ay(G) for some oy-hypergeometric G, clarifying the connection to the additive version of Hilbert’s Theorem 90.

Background

Throughout the paper the authors paper summability and creative telescoping for (q-)hypergeometric terms in a unified framework, assuming in most results that the q-parameter is not a root of unity. They explicitly remark that the case of q being a root of unity introduces additional algebraic complications.

They point out that, in this special case, the summability problem is closely related to the additive version of Hilbert’s Theorem 90. Extending their reduction-based framework and summability criteria to handle q being a root of unity would broaden applicability and illuminate the algebraic structure underlying q-differences in periodic settings.