A $q$-Clausen-Orr type formula and its applications

We show that certain terminating ${}_6{\varphi }_5$ series can be factorized into a product of two ${}_3{\varphi }_2$ series. As applications we prove a summation formula for a product of two $q$-Delannoy numbers along with some congruences for sums involving $q$-Delannoy numbers. This confirms three recent conjectures of the second author.