Q-Operator Identities And Its Applications

The history of q—series has been more than two hundreds and kinds of ways have been used to study it. q—Operator method seems to be one of the best methods. Mathematicians like L. Euler, L. J. Rogers, G.-C. Rota, S. Roman, J. Cigler, M. E. H. Ismail, G. E. Andrews and R. Askey have employed q—operator method in studying the q—series. In this article, we continue to apply q-operator method to study the theory of q-series.We start from q—commutative binomial theorem, the q—Leibniz rule and q—forward operator identity can be arrived. Using these two identities, we can get some q—identities and q—transformation formulas. Finally, we construct two q-commutative operators and use q—commutative operator identity, a nice q-identities can be obtained. This identity includes q-binomial theorem, Jacobi triple-product identity and the key identity for proving the quintuple-product identity.