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 hn (x, y | q). We introduce a q-difference operator θxy 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 sn (x, y ; b | q) and derive their generating function by using the new homogeneous q-shift operator L (b θxy). © 2009 Elsevier Inc. All rights reserved.
Saad, H. L., & Sukhi, A. A. (2010). Another homogeneous q-difference operator. Applied Mathematics and Computation, 215(12), 4332–4339. https://doi.org/10.1016/j.amc.2009.12.061