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

TL;DR

This paper presents a factorization of specific terminating basic hypergeometric series and applies it to prove new summation formulas and congruences involving $q$-Delannoy numbers, confirming recent conjectures.

Contribution

It introduces a novel factorization of $_{6}phi_5$ series into $_{3}phi_2$ series and applies this to derive new identities and prove conjectures related to $q$-Delannoy numbers.

Findings

Factorization of $_{6}phi_5$ series into $_{3}phi_2$ series

Summation formula for a product of two $q$-Delannoy numbers

Confirmed three recent conjectures involving $q$-Delannoy numbers

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.