Applications of two summation theorems of Gosper for the ${}_5F_4$ hypergeometric series

By means of two summation theorems of R. W. Gosper for a terminating ${}_5F_4$ hypergeometric series of arguments $1/4$ and $4$, we derive two general double-series identities involving a bounded sequence of arbitrary complex numbers. These series are then applied to obtain two reduction formulas for the Srivastava–Daoust double hypergeometric function.