The q-Borel Sum of Divergent Basic Hypergeometric Series r φ s (a;b;q,x)
Shunya Adachi · Symmetry Integrability and Geometry Methods and Applications · 2019
We study the divergent basic hypergeometric series which is a q-analog of divergent hypergeometric series.This series formally satisfies the linear q-difference equation.In this paper, for that equation, we give an actual solution which admits basic hypergeometric series as a q-Gevrey asymptotic expansion.Such an actual solution is obtained by using q-Borel summability, which is a q-analog of Borel summability.Our result shows a q-analog of the Stokes phenomenon.Additionally, we show that letting q → 1 in our result gives the Borel sum of classical hypergeometric series.The same problem was already considered by Dreyfus, but we note that our result is remarkably different from his one.