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

Abstract: We show that certain terminating ${6}\phi_5$ series can be factorized into a product of two ${3}\phi_{2}$ 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.