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 $q^{\pm1}$-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.