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

We use a new $q$-exponential operator based on the $q^{\pm 1}$-derivative ${\text{D}}_{q^{\pm 1}}$ 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.