  Following a previous article we continue our study on non-terminating
hypergeometric series with one free parameter, which aims to find arithmetical
constraints for a given hypergeometric series to admit a gamma product formula.
In this article we exploit the concepts of duality and reciprocity not only to
extend already obtained results to a larger region but also to strengthen
themselves substantially. Among other things we are able to settle the
rationality and finiteness conjectures posed in the previous article.
