Polynomials and asymptotic constants in a resurgent problem from ’t Hooft

Abstract

In a recent study of the quantum theory of harmonic oscillators, Gerard ’t Hooft proposed the following problem: given \(G(z)=\sum _{n=1}^\infty \sqrt{n}\,z^n\) for \(|z|<1\), find its analytic continuation for \(|z|\ge 1\), excluding a branch-cut \(z\in [1,\,\infty )\). A solution is provided by the bilateral convergent sum \(G(z)=\frac{1}{2}\sqrt{\pi }\sum _{n=-\infty }^\infty (2\pi \textrm{i}n-\log (z))^{-3/2}\). On the negative real axis, \(G(-\textrm{e}^u)\) has a sign-constant asymptotic expansion in \(1/u^2\), for large positive u. Optimal truncation leaves exponentially suppressed terms in an asymptotic expansion \(\textrm{e}^{-u}\sum _{k=0}^\infty P_k(x)/u^k\), with \(P_0(x)=x-\frac{2}{3}\) and \(P_k(x)\) of degree \(2k+1\) evaluated at \(x=u/2-\lfloor u/2\rfloor \). At large k, these polynomials become excellent approximations to sinusoids. The amplitude of \(P_k(x)\) increases factorially with k and its phase increases linearly, with \(P_k(x)\sim \sin ((2k+1)C-2\pi x) R^{2k+1}\Gamma (k+\frac{1}{2})/\sqrt{2\pi }\), where \(C\approx 1.0688539158679530121571\) and \(R\approx 0.5181839789815558726739\) are asymptotic constants satisfying \(R\exp (\textrm{i}\,C)=\sqrt{-1/(2+\pi \textrm{i})}.\)