Certain generating functions for Cigler's polynomials

Abstract

In this paper, we use the homogeneous $q$-operators [J. Difference Equ. Appl. {\bf20 } (2014), 837--851.] to derive Rogers formulas, extended Rogers formulas and Srivastava-Agarwal type bilinear generating functions for Cigler's polynomials [J. Difference Equ. Appl. {\bf 24} (2018), 479--502.]. Finally, we also derive two interesting transformation formulas between ${}_2\Phi_1, \, {}_2\Phi_2$ and ${}_3\Phi_2$.