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

arXiv:2509.26297v3 Announce Type: replace-cross 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 $R\exp({\rm i}\,C)=\sqrt{-1/(2+\pi{\rm i})}$.

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 $R\exp({\rm i}\,C)=\sqrt{-1/(2+\pi{\rm i})}$.

14 pages, 1 figure. Accepted for publication in Letters in Mathematical Physics. The exposition was improved based on the referees' comments

Number Theory (math.NT); High Energy Physics - Theory (hep-th); Quantum Physics (quant-ph)

arXiv:2509.26297 [math.NT]

(or arXiv:2509.26297v3 [math.NT] for this version)

https://doi.org/10.48550/arXiv.2509.26297 Focus to learn more arXiv-issued DOI via DataCite

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

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