Generalized Cigler’s Polynomials with Generalized Homogeneous \(q\)-Shift Operator

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DOI: https://doi.org/10.28924/2291-8639-24-2026-219
Published: Jul 17, 2026
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Ahmed F. Jaber, Husam L. Saad, Generalized Cigler’s Polynomials with Generalized Homogeneous \(q\)-Shift Operator, Int. J. Anal. Appl., 24 (2026), 219.

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Ahmed F. Jaber, Husam L. Saad

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 \(\Psi^{(\textbf{a};\textbf{b})}_n(x,y,z|q)\).

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