q-Derivative Operators and Basic Hypergeometric Series

By means of q-derivative operators, we investigate formal power series expansions. Two main expansion formulae in terms of q-derivative operators are established which can be considered as extensions of the corresponding results due to Carlitz (1973) and Liu (2002). Their applications to basic hypergeometric evaluations and transformations are discussed through series compositions and their q-derivative operations. Direct verification of the two main theorems are also presented.