Certain generating functions for Cigler's polynomials

TL;DR

This paper derives new generating functions and transformation formulas for Cigler's polynomials using homogeneous q-operators, expanding the analytical tools available for these polynomials.

Contribution

It introduces Rogers, extended Rogers, and bilinear generating functions for Cigler's polynomials, along with new transformation formulas between basic hypergeometric functions.

Findings

Derived Rogers formulas for Cigler's polynomials

Established extended Rogers and bilinear generating functions

Presented transformation formulas between hypergeometric functions

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 ${}_3\Phi_2$.