Exponentially-improved asymptotics for q-difference equations: ₂φ₀ and q PI

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 nth coefficient is often of the size q-12 n(n-1), in which q∈(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 ₂φ₀ and for solutions of the q-difference first Painlev\'e equation q PI. These are optimal truncated expansions, and re-expansions in terms of new q-hyperterminant functions. The re-expansions do incorporate the Stokes phenomena.