Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A $q$-Exponential Operator Based on the Derivative of Order 1 and Summation of Bilateral Basic Hypergeometric Series

Ronald Orozco López

TL;DR

The paper introduces a novel $q$-exponential operator built from the $q^{\pm1}$-derivative of order 1 to derive summation formulas for bilateral basic hypergeometric series. It develops a robust calculus of order 1 and a parameter-augmented operator $\mathrm{E}_{q}(y\mathbf{D}_{q^{\pm1}}|q^b)$, then applies this framework to obtain closed forms for ${}_{0}\psi_{1}$, ${}_{1}\psi_{1}$, ${}_{1}\psi_{2}$, and ${}_{2}\psi_{2}$, including bilateral series whose terms are basic hypergeometric functions. The results unify Jacobi theta function identities with Ramanujan’s and Bailey-Daum summations, delivering a versatile toolbox for evaluating bilateral $q$-series. This work advances the toolkit for $q$-series and has potential implications for modular forms, combinatorics, and related areas of mathematical analysis.

Abstract

We use a new $q$-exponential operator based on the $q^{\pm1}$-derivative $\D_{q^{\pm1}}$ of order 1 to derive summation formulas for bilateral basic hypergeometric series ${}_{0}ψ_{1}$, ${}_{1}ψ_{1}$, ${}_{1}ψ_{2}$, and ${}_{2}ψ_{2}$. In addition, we provide summation formulas for bilateral series whose terms are basic hypergeometric functions.

A $q$-Exponential Operator Based on the Derivative of Order 1 and Summation of Bilateral Basic Hypergeometric Series

TL;DR

The paper introduces a novel -exponential operator built from the -derivative of order 1 to derive summation formulas for bilateral basic hypergeometric series. It develops a robust calculus of order 1 and a parameter-augmented operator , then applies this framework to obtain closed forms for , , , and , including bilateral series whose terms are basic hypergeometric functions. The results unify Jacobi theta function identities with Ramanujan’s and Bailey-Daum summations, delivering a versatile toolbox for evaluating bilateral -series. This work advances the toolkit for -series and has potential implications for modular forms, combinatorics, and related areas of mathematical analysis.

Abstract

We use a new -exponential operator based on the -derivative of order 1 to derive summation formulas for bilateral basic hypergeometric series , , , and . In addition, we provide summation formulas for bilateral series whose terms are basic hypergeometric functions.

Paper Structure

This paper contains 11 sections, 58 theorems, 168 equations.

Table of Contents

  1. Introduction
  2. Calculus of order 1
  3. The $\lambda$-derivative of order 1 of a function
  4. Some $q^{\pm1}$-derivative identities
  5. Derivative of order 1 of the Jacobi theta function
  6. The $q$-operator $\mathrm{E}_{q}(y\mathbf{D}_{q^{\pm1}}|q^b)$
  7. Summation formulas involving Jacobi's theta function
  8. Application to bilateral basic hypergeometric series
  9. Summations involving Jacobi theta function
  10. Summations involving Ramanujan summation
  11. Summations involving Bailey-Daum summation formula

Key Result

Theorem 1

For all $\alpha,\beta,\gamma\in\mathbb{C}$,

Theorems & Definitions (98)

  • Definition 1
  • Theorem 1
  • Proposition 1: Product $\lambda$-rule
  • Proposition 2
  • Proposition 3: Quotient $\lambda$-rule
  • Proposition 4
  • proof
  • Example 1
  • Proposition 5: Leibniz $\lambda$-rule
  • proof
  • ...and 88 more