Another homogeneous q-difference operator

Abstract

In this paper we show the equivalence between Goldman–Rota q-binomial identity and its inverse. We may specialize the value of the parameters in the generating functions of Rogers–Szegö polynomials to obtain some classical results such as Euler identities and the relation between classical and homogeneous Rogers–Szegö polynomials. We give a new formula for the homogeneous Rogers–Szegö polynomials h n ( x , y | q ) . We introduce a q-difference operator θ xy on functions in two variables which turn out to be suitable for dealing with the homogeneous form of the q-binomial identity. By using this operator, we got the identity obtained by Chen et al. [W.Y.C. Chen, A.M. Fu, B. Zhang, The homogeneous q-difference operator, Advances in Applied Mathematics 31 (2003) 659–668, Eq. (2.10)] which they used it to derive many important identities. We also obtain the q-Leibniz formula for this operator. Finally, we introduce a new polynomials s n ( x , y ; b | q ) and derive their generating function by using the new homogeneous q-shift operator L ( b θ xy ) .