The `1/9'-problem for the φ-functions
Abstract
The functions φℓ(z)=(ez−sℓ(z))/zℓ, with sℓ the Taylor polynomial of ez of degree ℓ−1, are what exponential integrators evaluate, and they evaluate them through rational approximations on the negative real axis ℝ-. Schmelzer and Trefethen (2007/08) conjectured that the best such approximations converge at the rate of the classical `1/9'-problem: En(φℓ)1/n→ H, Halphen's constant, for every ℓ. We prove it. The theorem behind it gives the same rate for f=u0+u1exp with u0 rational and u1not≡0 meromorphic with finitely many poles and log|u1(z)|=o(|z|), the lower bound required only away from ℝ-, whenever f has no pole on ℝ- (Theorem~4.4). For rational u1 this is Theorem 1 of Stahl and Schmelzer (2009), announced with its proof deferred to a manuscript that never appeared; the hypotheses admit, for instance, u1(z)=sinh(π√z)/(π√z), whose zeros are infinite in number. The proof is a reading of Gonchar and Rakhmanov (1989) rather than an extension of them. Their theorem concerns a sequence given by a Cauchy-type integral with a varying weight Φn, and a contour identity of Schmelzer and Trefethen presents u1exp, once its principal parts are removed, in exactly that form, with Φn(t)=u1(−nt)e−nt. The factor u1(−nt) grows subexponentially in n, and their hypothesis on Φn divides by 2n, so it drops out of the external field: field, extremal arc, S-property and constant are all theirs, unchanged. For φℓ the factor is t−ℓ, whose pole sits at the origin, a point of the approximation set and hence at positive distance from every admissible contour.
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BibTeX
@misc{schmelzer202619problem,
title = {{The `1/9'-problem for the $φ$-functions}},
author = {Thomas Schmelzer},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14489},
eprint = {2609.14489},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14489},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The `1/9'-problem for the φ-functions
AU - Thomas Schmelzer
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.14489
UR - https://arxiv.org/abs/2609.14489
AB - The functions φ_ℓ(z)=(e^z−s_ℓ(z))/z^ℓ, with s_ℓ the Taylor polynomial of e^z of degree ℓ−1, are what exponential integrators evaluate, and they evaluate them through rational approximations on the negative real axis ℝ_-. Schmelzer and Trefethen (2007/08) conjectured that the best such approximations converge at the rate of the classical `1/9'-problem: E_n(φ_ℓ)^1/n→ H, Halphen's constant, for every ℓ. We prove it. The theorem behind it gives the same rate for f=u_0+u_1exp with u_0 rational and u_1not≡0 meromorphic with finitely many poles and log|u_1(z)|=o(|z|), the lower bound required only away from ℝ_-, whenever f has no pole on ℝ_- (Theorem~4.4). For rational u_1 this is Theorem 1 of Stahl and Schmelzer (2009), announced with its proof deferred to a manuscript that never appeared; the hypotheses admit, for instance, u_1(z)=sinh(π√z)/(π√z), whose zeros are infinite in number. The proof is a reading of Gonchar and Rakhmanov (1989) rather than an extension of them. Their theorem concerns a sequence given by a Cauchy-type integral with a varying weight Φ_n, and a contour identity of Schmelzer and Trefethen presents u_1exp, once its principal parts are removed, in exactly that form, with Φ_n(t)=u_1(−nt)e^−nt. The factor u_1(−nt) grows subexponentially in n, and their hypothesis on Φ_n divides by 2n, so it drops out of the external field: field, extremal arc, S-property and constant are all theirs, unchanged. For φ_ℓ the factor is t^−ℓ, whose pole sits at the origin, a point of the approximation set and hence at positive distance from every admissible contour.
ER -