Generalized homogeneous q-shift operator applied for generalized Cigler's polynomials

Husam L. Saad; Samaher A.  Abdul-Ghani

doi:10.5269/bspm.76720

Husam L. Saad University of Basrah
Samaher A. Abdul-Ghani https://orcid.org/0000-0001-5125-3399

Abstract

In this paper, we establish the generalized homogeneous $q$-shift operator $E(q^\alpha z\theta_{xy})$ and the generalised Cigler's polynomials $\mathbb{D}^{(\alpha-n)}_n(\sigma_1, \cdots, \sigma_{r},\rho_1,\cdots, \rho_s,x,y,z)$. Then, we apply this operator to derive some $q$-identities such as: the generating function and its extension, Rogers formula and its extension, Mehler's formula and its extension, Srivastava-Agarwal type bilinear generating functions to the polynomials $\mathbb{D}^{(\alpha-n)}_n(\sigma_1, \cdots, \sigma_{r},\rho_1,\cdots, \rho_s,x,y,z)$. In addition, we supply some special values for the identities of $\mathbb{D}^{(\alpha-n)}_n(\sigma_1, \cdots, \sigma_{r},\rho_1,\cdots, \rho_s,x,y,z)$ in order to establish the same identities for the polynomials $D^{(\alpha-n)}_{n}(x,y,b)$ and $\Psi_n^{(\textbf{a},\textbf{b})}(x,y,z|q)$.

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