The Homogeneous q-Difference Operator
William Y. C. Chen , Amy M. Fu and Baoyin Zhang
Abstract: We introduce a q-differential operator Dxy on functions in two variables which turns out to be suitable for dealing with the homogeneous form of the q-binomial theorem as studied by Andrews, Goldman and Rota, Roman, Ihrig and Ismail, et al. The homogeneous versions of the q-binomial theorem and the Cauchy identity are often useful for their specializations of the two parameters. Using this operator, we derive an equivalent form of the Goldman-Rota binomial identity and show that it is a homogeneous generalization of the q-Vandermonde identity. Moreover, the inverse identity of Goldman and Rota also follows from our unified identity. We also obtain the q-Leibniz formula for this operator. In the last section, we introduce the homogeneous Rogers-Szegö polynomials and derive their generating function by using the homogeneous q-shift operator.
Keywords: q-binomial theorem, Cauchy polynomials, q-Vandermonde identity, homogeneous q-difference operator, q-Leibniz formula, homogeneous Rogers-Szegö polynomials.