Polynomials and asymptotic constants in a resurgent problem from 't Hooft
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914087329333248 |
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| author | Broadhurst, David Nemes, Gergő |
| author_facet | Broadhurst, David Nemes, Gergő |
| contents | 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 $G(z)=\frac12\sqrtπ\sum_{n=-\infty}^\infty(2π{\rm i}n-\log(z))^{-3/2}$. 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 $P_k(x)\sim\sin((2k+1)C-2πx)R^{2k+1}Γ(k+\frac12)/\sqrt{2π}$, where $C\approx1.0688539158679530121571$ and $R\approx0.5181839789815558726739$ are asymptotic constants satisfying $R\exp({\rm i}\,C)=\sqrt{-1/(2+π{\rm i})}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_26297 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Polynomials and asymptotic constants in a resurgent problem from 't Hooft Broadhurst, David Nemes, Gergő Number Theory High Energy Physics - Theory Quantum Physics 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 $G(z)=\frac12\sqrtπ\sum_{n=-\infty}^\infty(2π{\rm i}n-\log(z))^{-3/2}$. 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 $P_k(x)\sim\sin((2k+1)C-2πx)R^{2k+1}Γ(k+\frac12)/\sqrt{2π}$, where $C\approx1.0688539158679530121571$ and $R\approx0.5181839789815558726739$ are asymptotic constants satisfying $R\exp({\rm i}\,C)=\sqrt{-1/(2+π{\rm i})}$. |
| title | Polynomials and asymptotic constants in a resurgent problem from 't Hooft |
| topic | Number Theory High Energy Physics - Theory Quantum Physics |
| url | https://arxiv.org/abs/2509.26297 |