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[16,17],L.J.Rogers[30-32], G.-C.Rota[36], S.Roman[33-35], J.Cigler[10-13], G.E.Andrews[1] and Chen and liu[7, 8] 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.In this paper,we first give the definition of q- operator,with the analysis of D_q , 9 operator,we can get some special formulas and transformation formulas between them .Secondly ,using these operator identities ,we can get some new proofs of q- series identities ,for example, q- Cauchy Theorem , q- Euler -identity and so on .Thirdly,on the foundation of q- Chu-Vandermonde,using D_q,θoperators,acting on the variable a,c ,we can get some new proofs of q - Pfaff - Saalchütz ,we also get some transformation formulas of 3φ2, 4φ3 and so on.