The q-Borel Sum of Divergent Basic Hypergeometric Series ᵣφₛ(a; b; q, x)

Abstract

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.