Two q -difference equations and q -operator identities

In this paper, we establish two general q-exponential operator identities by solving two simple q-difference equations, which contain two well-known operator identities as special cases. These operator identities allow us to derive naturally the q-Mehler formulas for the Rogers-Szegö polynomials and the q − 1-Rogers-Szegö polynomials. The q-Mehler formulas are used to derive two q-exponential operator identities involving series. We derive a q-integral formula which is an extension of the q-integral form of the Sears transformation. Finally, we set up a general identity for the bilateral q-series and from which a simple proof of Bailey's summation is given.