# Halphen's constant

## Definitions

Symbol: HalphenConstant $\Lambda$ Halphen's constant (one-ninth constant) 0.10765...

## Numerical value

$\Lambda = 0.10765391922648457661532344509094719058797656329012 \;\, {\scriptstyle (\text{nearest } 50 \text{ digits})}$
$\frac{1}{\Lambda} = 9.2890254919208189187554494359517450610316948677501 \;\, {\scriptstyle (\text{nearest } 50 \text{ digits})}$
$\Lambda \ne \frac{1}{9}$

## Approximation theory

$\Lambda = \lim_{n \to \infty} {\lambda}_{n}^{1 / n}\; \text{ where } R = \left\{ r : r \in \mathbb{R}(t) \,\mathbin{\operatorname{and}}\, \deg(r) \le \left(n, n\right) \right\},\;{\lambda}_{n} = \mathop{\operatorname{inf}}\limits_{r \in R} \mathop{\operatorname{sup}}\limits_{x \in \left(-\infty, 0\right]} \left|{e}^{x} - r(x)\right|$

## Formulas

$\Lambda = \mathop{\operatorname{zero*}\,}\limits_{x \in \left(0, 1\right)} \left(-\frac{1}{8} + \sum_{n=1}^{\infty} \frac{n {x}^{n}}{1 - {\left(-x\right)}^{n}}\right)$
$\Lambda = \mathop{\operatorname{zero*}\,}\limits_{x \in \left(0, 1\right)} \left[\sum_{n=0}^{\infty} {\left(2 n + 1\right)}^{2} {\left(-x\right)}^{n \left(n + 1\right) / 2}\right]$
$\Lambda = \mathop{\operatorname{zero*}\,}\limits_{x \in \left(0, 1\right)} \left(-\frac{1}{8} + \sum_{n=1}^{\infty} \left|\sum_{d \mid n} {\left(-1\right)}^{d} d\right| {x}^{n}\right)$
$\Lambda = \exp\!\left(-\frac{\pi K\!\left(1 - c\right)}{K(c)}\right)\; \text{ where } c = \mathop{\operatorname{zero*}\,}\limits_{m \in \left(0, 1\right)} \left(K(m) - 2 E(m)\right)$
$\Lambda = \mathop{\operatorname{zero*}\,}\limits_{x \in \left(0, 1\right)} \left[\theta''_{2}\!\left(0 , \frac{\log\!\left(-x\right)}{2 \pi i}\right)\right]$

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2020-01-31 18:09:28.494564 UTC