Terminating balanced \({}_4\phi_3\)-series and very well-poised \({}_8\phi_7\)-series (Q2320052)

The goal of the authors is to investigate the following terminating balenced series: \[ \Omega_{\lambda, \rho}^{n}(b, d)= {}_{4}\phi_{3}\left[\left. \begin{array}{r} q^{-n}, q^{\lambda} b, q^{1 / 2} b, q^{n+\rho} d^{2} \\ qd, q^{1 / 2} d, q^{\lambda+\rho} b^{2}\end{array} \right| q; q\right]. \] The parameters \(\lambda\) and \(\rho\) are integer, the other two parameters, \(b\) and \(d\) are arbitrary indeterminates. It is shown that one can annihilate one of the parameters via two special reduction formulas. Many of the \(\Omega_{\lambda, \rho}^{n}(b, d)\) sums can be evaluated explicitly, as the authors show. A large number of examples can be found at the end of the paper which show the applicability of the results.