Exponentially-improved asymptotics for $q$-difference equations: ${}_2{\varphi }_0$ and $q{\mathrm{P}}_{\mathrm{I}}$

Abstract

Usually when solving differential or difference equations via series solutions one encounters divergent series in which the coefficients grow like a factorial. Surprisingly, in the $q$-world the $n$th coefficient is often of the size $q^{-\frac{1}{2}n(n-1)}$, in which $q\in (0,1)$ is fixed. Hence, the divergence is much stronger, and one has to introduce alternative Borel and Laplace transforms to make sense of these formal series. We will discuss exponentially-improved asymptotics for the basic hypergeometric function ${}_2{\varphi }_0$ and for solutions of the $q$-difference first Painlevé equation $q{\mathrm{P}}_{\mathrm{I}}$. These are optimal truncated expansions, and re-expansions in terms of new $q$-hyperterminant functions. The re-expansions do incorporate the Stokes phenomena.

15 pages, 1 figure