Generalized Cigler’s Polynomials with Generalized Homogeneous $q$-Shift Operator

Jaber, Ahmed F.; Saad, Husam L.

doi:10.28924/2291-8639-24-2026-219

Research Article

Generalized Cigler’s Polynomials with Generalized Homogeneous $q$-Shift Operator

Ahmed F. Jaber
Husam L. Saad*

* Corresponding author

Abstract

This study employs the generalized homogeneous $q$-shift operator, defined as $\displaystyle\ _{r+1}\Phi_s \left(\begin{array}{c} q^{-\alpha},\alpha_1, \cdots, \alpha_{r}\\ \beta_1,\cdots, \beta_s\\ \end{array}; q,q^\alpha \rho D_{\chi\psi} \right)$. Subsequently, by utilizing this operator, we derive several polynomials' $q$-identities. These derived identities include the generating function, the Rogers' formula, Mehler's formula along with its extension, and three distinct mixed generating functions for the generalized Cigler's polynomials $\mathcal{C}^{(\alpha-n)}_{n}(\boldsymbol{\alpha}, \boldsymbol{\beta},\chi,\psi,\rho|q)$. Furthermore, we establish a transformational identity linking various generating functions specifically for $\mathcal{C}^{(\alpha-n)}_n(\boldsymbol{\alpha}, \boldsymbol{\beta},\chi,\psi,\rho|q)$. Finally, by proposing certain specific values for $\mathcal{C}^{(\alpha-n)}_n(\boldsymbol{\alpha}, \boldsymbol{\beta},\chi,\psi,\rho|q)$, we establish new identities for the extended Verma-Jain polynomials $V^{(\textbf{a};\textbf{c})}_n(x,y,z|q)$ and a related class of generalized $q$-hypergeometric polynomials $\Psi^{(\textbf{a};\textbf{b})}_n(x,y,z|q)$.

How to Cite

A. F. Jaber, H. L. Saad, Generalized Cigler’s Polynomials with Generalized Homogeneous $q$-Shift Operator, Int. J. Anal. Appl. 24 (2026), 219. https://doi.org/10.28924/2291-8639-24-2026-219

BibTeX

@article{Jaber2026,
  author  = {Jaber, Ahmed F. and Saad, Husam L.},
  title   = {Generalized Cigler’s Polynomials with Generalized Homogeneous \(q\)-Shift Operator},
  journal = {Int. J. Anal. Appl.},
  volume  = {24},
  year    = {2026},
  pages   = {219},
  doi     = {10.28924/2291-8639-24-2026-219}
}

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