q-Difference equation and the Cauchy operator identities

In this paper, we verify the Cauchy operator identities by a new method. And by using the Cauchy operator identities, we obtain a generating function for Rogers–Szegö polynomials. Applying the technique of parameter augmentation to two multiple generalizations of q-Chu–Vandermonde summation theorem given by Milne, we also obtain two multiple generalizations of the Kalnins–Miller transformation.