The $q$-Laplace Transforms compared: the basic confluent hypergeometric function ${}_2{\phi }_0$

In solving $q$-difference equations, and in the definition of $q$-special functions, we encounter formal power series in which the $n$th coefficient is of size $q^{-\left(\genfrac{}{}{0px}{}{n}{2}\right)}$ with $q\in (0,1)$ fixed. To make sense of these formal series, a $q$-Borel-Laplace resummation is required. There are three candidates for the $q$-Laplace transform, resulting in three different resummations. Surprisingly, the differences between these resummations have hardly been discussed in the literature. Our main result provides explicit formulas for these $q$-exponentially small differences. We also give simple Mellin–Barnes integral representations for all the basic hypergeometric ${}_r{\phi }_s$ functions and derive a third (discrete) orthogonality condition for the Stieltjes–Wigert polynomials. As the main application, we introduce three resummations for the ${}_2{\phi }_0$ functions which can be seen as $q$ versions of the Kummer $U$ functions. We derive many of their properties, including interesting integral and sum representations, connection formulas, and error bounds.