Efficient numerical computation of the q-hyperterminant function

Develop an efficient numerical method to compute the q-hyperterminant function F_q(z; N; σ) across the full complex plane, where F_q(z; N; σ) is defined by F_q(z; N; σ)=σ^{1−N} C_q ∫_0^{∞} [t^{N−1} E_q(t)]/(t+σ z) dt/t with E_q(t)=exp((ln(t/√q))^2/(2 ln q)) and C_q=(−2π ln q)^{−1/2}.

Background

The paper introduces the q-hyperterminant F_q(z; N; σ) as a q-analogue of classical hyperterminants, designed to capture the q-Stokes phenomenon in exponentially improved asymptotics. It serves as a principal building block for re-expansions of optimal truncated asymptotic series for both basic hypergeometric functions (e.g., {}_2φ_0) and solutions of the q-difference first Painlevé equation.

In Section 4, the authors derive many properties of F_q, including recurrence relations, reflection formulas, the Stokes phenomenon, and a full uniform asymptotic expansion across Stokes curves showing Berry smoothing. Despite these analytical advances, an efficient numerical method to compute F_q over the entire complex plane remains unaddressed and is explicitly posed as an open problem.

References

Two open problems are the following. (1) For the $q$-hyperterminant we give in \S\ref{Sect:qhyperterminant} many of its properties and even a full uniform asymptotic expansion across the Stokes curve, but we still need an efficient method to numerically compute this function in the full complex plane.

— Exponentially-improved asymptotics for $q$-difference equations: ${}_2φ_0$ and $q{\rm P}_{\rm I}$ (2403.02196 - Joshi et al., 4 Mar 2024) in Section 1 (Introduction and summary)