The homogeneous q-difference operator |
| |
Authors: | William Y. C. Chen Amy M. Fu Baoyin Zhang |
| |
Affiliation: | Center for Combinatorics, LPMC, Nankai University, Tianjin 300071, PR, China |
| |
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 |
本文献已被 ScienceDirect 等数据库收录! |
|