Parameter augmentation and the q-Gosper algorithm

We develop a method for deriving new basic hypergeometric identities from old ones by parameter augmentation. The main idea is to introduce a new parameter and use the q-Gosper algorithm to find out a suitable form of the summand. By this method, we recover some classical formulas on basic hypergeometric series and find extensions of the Rogers–Fine identity and Ramanujan’s summation formula. Moreover, we derive an identity for a summation.