An improvement of the two-line algorithm for proving \(q\)-hypergeometric identities. (Q1398270)

The idea that one can prove a hypergeometric identity by checking a finite number of special cases was known to \textit{D. Zeilberger} [J. Math. Anal. Appl. 85, 114--145 (1982; Zbl 0485.05003)]. \textit{L. Yen} [J. Math. Anal. Appl. 213, 1--14 (1997; Zbl 0903.33008)] gave a two-line algorithm to show that \(q\)-hypergeometric identities \(\sum_kF(n,k)=1\); \(n\geq n_0\); can be proved by checking that they are correct for \(n\in\{n_0,n_0 +1,\dots, n_1\}\), where \(n_1\) is a polynomial of degree of 24 in the parameters of \(F(n,k)\). In this paper, the author improves this result by giving a specific formula for \(n_1\), bounded above by a polynomial of degree 9 in the parameters of \(F(n,k)\). As illustration, he applies the obtained result to the \(q\)-Vandermonde-Chu identity and to the Jacobi's triple product identity.