Bibliographic Details
| Title: |
The homogeneous q-difference operator |
| Authors: |
Chen, William Y.C. chenstation@yahoo.com, Fu, Amy M.1 fmu@eyou.com, Zhang, Baoyin1 zby75@eyou.com |
| Source: |
Advances in Applied Mathematics. Nov2003, Vol. 31 Issue 4, p659. 10p. |
| Subjects: |
Differential operators, Binomial theorem, Polynomials, Operator equations |
| 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–Szego¨ polynomials and derive their generating function by using the homogeneous q-shift operator. [Copyright &y& Elsevier] |
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| Database: |
Engineering Source |
