Parameter augmentation and the \(q\)-Gosper algorithm (Q999087)

Given a \(q\)--hypergeometric term \(A_k\) (i.e., \(A_{k+1}/A_k\) can be represented by a rational function in \(q^k\)), the \(q\)--Gosper algorithm decides in finitely many steps, if there exists a \(q\)-hypergeometric term \(g_k\) such that the telescoping equation \(g_{k+1}-g_k=A_k\) holds; in case of existence, it computes such a \(g_k\). In this article, the author follows Zeilberger's creative telescoping paradigm based on \(q\)-Gosper's algorithm to search for a recurrence relation in terms of a discrete parameter \(n\). In addition, an extra parameter \(f\) is involved that arises in a certain \(q\)--shifted factorial of \(A_k\), and the author hunts for particular values for \(f\) such that the found recurrence is simple (ideally, the recurrence should have order \(0\), i.e., \(A_k\) should be Gosper-summable). In this way, various classical basic hypergeometric identities and non-trivial extensions of them can be found.