What question did this study set out to answer?

The aim is to introduce and explore deformed homogeneous polynomials and their relationship with classical polynomials and operators.

February 12, 2026Open Access

Deformed Homogeneous Polynomials and the Generalized q-Exponential Operator

Key Points

The aim is to introduce and explore deformed homogeneous polynomials and their relationship with classical polynomials and operators.
Introduced $ ext{R}_{n}(x,y;u|q)$ as a generalization of classical polynomials.
Derived basic properties, recurrence relations, and $q$-difference equations.
Defined the $u$-deformed $q$-exponential operator $ ext{E}(yD_{q}|u)$.
Provided generating functions including generalizations of known formulas.
Successfully derived new transformation formulas for basic hypergeometric series.
Generated relationships with known operators such as $ ext{T}(yD_{q})$ and $ ext{R}(yD_{q})$ for specific values of $u$ and $q$.
Developed generalized generating functions for $ ext{R}_{n}(x,y;u|q)$.

Abstract

In this paper, we introduce the deformed homogeneous polynomials R₍ (x, y;u|q). These polynomials generalize some classical polynomials: the Rogers-Szegö polynomials h₍ (x|q), the generalized Rogers-Szegö polynomials r₍ (x, y), the Stieltjes-Wigert polynomials S₍ (x;q), among others. Basic properties of the polynomial R₍ are given, along with recurrence relations, its q-difference equation, and representations. Generating functions for the polynomials R₍ (x, y;u|q) are given. These functions include generalizations of the Mehler and Rogers formulas. In addition, generalizations of the q-binomial formula and the Heine transformation formula are obtained. These results are obtained via the u-deformed q-exponential operator E (yDₐ|u), defined here. From this operator, we obtain for free the operators T (yDₐ) the Chen, R (yDₐ) of Saad, E (yDₐ) of Exton, and R (yDₐ) of Rogers-Ramanujan when u=1, q, q, q², respectively. We introduce the deformed basic hypergeometric series ₑΦₒ, a generalization of the classical basic hypergeometric series. New transformation formulas for basic hypergeometric series are obtained.

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