The Generalized Homogeneous q-Shift Operator  _r Φ_s (D_xy ) for q-Identities and q-Integrals

Samaher A. Abdul-Ghani; Husam L. Saad

doi:10.24996/ijs.2023.64.11.29

Authors

Samaher A. Abdul-Ghani Department of Mathematics, College of Science, Basrah University, Basrah, Iraq https://orcid.org/0000-0001-5125-3399
Husam L. Saad Department of Mathematics, College of Science, Basrah University, Basrah, Iraq

DOI:

https://doi.org/10.24996/ijs.2023.64.11.29

Keywords:

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

Abstract

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.