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

TL;DR

This paper introduces a new $q$-exponential operator based on the $q^{ ext{±1}}$-derivative of order 1, deriving summation formulas for various bilateral basic hypergeometric series and related functions.

Contribution

It presents a novel $q$-exponential operator and applies it to derive new summation formulas for bilateral basic hypergeometric series.

Findings

Derived summation formulas for ${}_{0} ilde{ ext{}}\psi_{1}$, ${}_{1} ilde{ ext{}}\psi_{1}$, ${}_{1} ilde{ ext{}}\psi_{2}$, and ${}_{2} ilde{ ext{}}\psi_{2}$ series.

Provided summation formulas for bilateral series with basic hypergeometric function terms.

Introduced a new operator that simplifies the derivation of hypergeometric series identities.

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 ${}_{0}\psi_{1}$, ${}_{1}\psi_{1}$, ${}_{1}\psi_{2}$, and ${}_{2}\psi_{2}$. In addition, we provide summation formulas for bilateral series whose terms are basic hypergeometric functions.