Extensions of two $q$-Gosper identities with an extra integer parameter

Abstract

According to the method of series rearrangement, we establish the extensions of two $q$-Gosper identities with an extra integer parameter. The limiting cases of them produce the generalizations of Gosper's two ${}_3{F}_2(\frac{3}{4})$-series identities with an additional integer parameter. Meanwhile, several related results are also given.