The Generalized Homogeneous q-Shift Operator _r Φ_s (D_xy ) for q-Identities and q-Integrals
homogeneous q-shift operator
Hiene's transformation
Jackson's transformation
q-hypergeometric polynomials
generating function
Rogers formula
Srivastava-Agarwal generating function
Askey-Wilson integral
Andrews-Askey q-integral
In this paper, we illustrate how to use the generalized homogeneous -shift operator in generalizing various well-known q-identities, such as Hiene's transformation, the q-Gauss sum, and Jackson's transfor- mation. For the polynomials , we provide another formula for the generating function, the Rogers formula, and the bilinear generating function of the Srivastava-Agarwal type. In addition, we also generalize the extension of both the Askey-Wilson integral and the Andrews-Askey integral.
