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

• URL: http://arxiv.org/abs/2509.26297v2
• Date: Sat, 11 Oct 2025 14:27:45 GMT
• Title: Polynomials and asymptotic constants in a resurgent problem from 't Hooft
• Authors: David Broadhurst, Gergő Nemes,
• Abstract summary: Given $G(z)=sum_n=1inftysqrtn,zn$ for $|z|1$, find its analytic continuation for $|z|ge1$, excluding a branch-cut $zin[1,,infty)$.<n>A solution is provided by the bilateral convergent sum $G(z)=frac12sqrtpisum_n=inftyinfty (2pirm in-log(z))-3/2
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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 $|z|\ge1$, excluding a branch-cut $z\in[1,\,\infty)$. A solution is provided by the bilateral convergent sum $G(z)=\frac12\sqrt{\pi}\sum_{n=-\infty}^\infty(2\pi{\rm i}n-\log(z))^{-3/2}$. On the negative real axis, $G(-{\rm e}^u)$ has a sign-constant asymptotic expansion in $1/u^2$, for large positive $u$. Optimal truncation leaves exponentially suppressed terms in an asymptotic expansion ${\rm e}^{-u}\sum_{k=0}^\infty P_k(x)/u^k$, with $P_0(x)=x-\frac23$ 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 $P_k(x)\sim\sin((2k+1)C-2\pi x)R^{2k+1}\Gamma(k+\frac12)/\sqrt{2\pi}$, where $C\approx1.0688539158679530121571$ and $R\approx0.5181839789815558726739$ are asymptotic constants satisfying $R\exp({\rm i}\,C)=\sqrt{-1/(2+\pi{\rm i})}$.

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