Applicability of the \(q\)-analogue of Zeilberger's algorithm (Q2456539)

Given a (\(q\)-)hypergeometric term \(T(n,k)\) in \(n\) and \(k\) and a nonnegative integer \(d\), Zeilberger's algorithm and its \(q\)-analogue version try to compute a creative telescoping solution of order \(d\). Given such a solution, one obtains, with some mild extra conditions, a recurrence relation for \(S(n)=\sum_kT(n,k)\). Here one usually starts looking for a solution of order \(0\) and, if this fails, one increases the possible order incrementally. In [\textit{S. A. Abramov}, Adv. Appl. Math. 30 No. 3, 424--441 (2003; Zbl 1030.33011)] a necessary and sufficient condition for the termination of this strategy has been provided. In other words, before trying to compute a creative telescoping solution of order \(d=0,1,2,\dots\), one can check as a preprocessing step, if there exists such such a solution for some large enough \(d\). In this article the authors can carry over the ideas from the hypergeometric to the \(q\)-hyper\-geometric case. Namely, in Theorem~4.6 they derive a necessary and sufficient condition of the following form. First compute (in a non-trivial way) a certain type of additive decomposition for the summand \(T(n,k)\). Then one can pick out a particular polynomial \(v(n,k)\) with the following property: Zeilberger's algorithms will succeed to compute a creative telescoping solution for some order \(d\) if and only if \(v(n,k)\) is \(q\)-proper.