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Deformed Homogeneous Polynomials and the Generalized $q$-Exponential Operator

Ronald Orozco López

TL;DR

The paper develops a unified framework of deformed polynomials and hypergeometric series built around a deformed $q$-exponential, centering on the two-parameter family $R_n(x,y;u|q)$ and its specializations to Rogers–Szegö, Stieltjes–Wigert, and Exton polynomials. It introduces the $u$-deformed $q$-exponential operator $\mathop{\hbox{E}}(yD_q|u)$, which directly generates $R_n$ and yields a suite of operator identities linking these polynomials to various deformed hypergeometric objects. The work then develops a comprehensive set of generating functions, transformation formulas, Mehler-type and Rogers-type formulas, and generalized ${}_{r}\Phi_{s}$ series, providing generalized $q$-binomial theorems and Heine transformations with deformations controlled by $u$ (and related parameters). By deriving $q$-difference equations and hypergeometric representations, the paper unifies and extends classical results for Rogers–Szegö, Saad’s generalizations, Srivastava–Agarwal-type formulas, and Rogers–Ramanujan constructions, with potential applications to related orthogonal polynomials and $q$-series identities.

Abstract

This paper introduces the deformed homogeneous polynomials $\mathrm{R}_{n}(x,y;u|q)$. These polynomials generalize some classical polynomials: the Rogers-Szegö polynomials $\mathrm{h}_{n}(x|q)$, the generalized Rogers-Szegö polynomials $\mathrm{r}_{n}(x,y)$, the Stieljes-Wigert polynomials $\mathrm{S}_{n}(x;q)$, among others. Basic properties of the polynomial $\mathrm{R}_{n}$ are given, along with recurrence relations, its $q$-difference equation, and representations. Generating functions for the polynomials $\mathrm{R}_{n}(x,y;u|q)$ 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 $\mathrm{E}(yD_{q}|u)$, defined here. From this operator, we obtain for free the operators T$(yD_{q})$ the Chen, $\mathrm{R}(yD_{q})$ of Saad, $\mathcal{E}(yD_{q})$ of Exton, and $\mathcal{R}(yD_{q})$ of Rogers-Ramanujan when $u=1,q,\sqrt{q},q^2$, respectively. We introduce the deformed basic hypergeometric series ${}_{r}Φ_{s}$, a generalization of the classical basic hypergeometric series. New transformation formulas for basic hypergeometric series are obtained.

Deformed Homogeneous Polynomials and the Generalized $q$-Exponential Operator

TL;DR

The paper develops a unified framework of deformed polynomials and hypergeometric series built around a deformed -exponential, centering on the two-parameter family and its specializations to Rogers–Szegö, Stieltjes–Wigert, and Exton polynomials. It introduces the -deformed -exponential operator , which directly generates and yields a suite of operator identities linking these polynomials to various deformed hypergeometric objects. The work then develops a comprehensive set of generating functions, transformation formulas, Mehler-type and Rogers-type formulas, and generalized series, providing generalized -binomial theorems and Heine transformations with deformations controlled by (and related parameters). By deriving -difference equations and hypergeometric representations, the paper unifies and extends classical results for Rogers–Szegö, Saad’s generalizations, Srivastava–Agarwal-type formulas, and Rogers–Ramanujan constructions, with potential applications to related orthogonal polynomials and -series identities.

Abstract

This paper introduces the deformed homogeneous polynomials . These polynomials generalize some classical polynomials: the Rogers-Szegö polynomials , the generalized Rogers-Szegö polynomials , the Stieljes-Wigert polynomials , among others. Basic properties of the polynomial are given, along with recurrence relations, its -difference equation, and representations. Generating functions for the polynomials are given. These functions include generalizations of the Mehler and Rogers formulas. In addition, generalizations of the -binomial formula and the Heine transformation formula are obtained. These results are obtained via the -deformed -exponential operator , defined here. From this operator, we obtain for free the operators T the Chen, of Saad, of Exton, and of Rogers-Ramanujan when , respectively. We introduce the deformed basic hypergeometric series , a generalization of the classical basic hypergeometric series. New transformation formulas for basic hypergeometric series are obtained.
Paper Structure (22 sections, 40 theorems, 125 equations)

This paper contains 22 sections, 40 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. $q$-differential operator
  3. Deformed basic hypergeometric series
  4. Deformed homogeneous polynomials
  5. Definition and basic properties
  6. Recurrence relations
  7. Deformed $q$-exponential function as limit of deformed homogeneous polynomials
  8. Deformed basic hypergeometric representation
  9. $q$-Difference equation
  10. The deformed $q$-exponential operator
  11. Generating functions for $\mathrm{R}_{n}(x,y;u|q)$
  12. Generalized $q$-binomial theorem
  13. Generalization of Heine's transformation formula
  14. Mehler's formula for $\mathrm{R}_{n}(x,y;u|q)$
  15. Srivastava-Agarwal type formulas
  16. ...and 7 more sections

Key Result

Theorem 1

The ${}_{r}\Phi_{s}$-series is an entire function if one of the following conditions is satisfied The ${}_{r}\Phi_{s}$-series converges in $\vert z\vert<1$ if $\vert u\vert=1$ and $1+s-r=0$. The ${}_{r}\Phi_{s}$-series converges for $z=0$ if $\vert u\vert>1$ and $1+s-r=0$.

Theorems & Definitions (65)

  • Definition 1
  • Theorem 1
  • Theorem 2
  • proof
  • Definition 2
  • Theorem 3
  • proof
  • Theorem 4
  • Theorem 5
  • proof
  • ...and 55 more