# Stieltjes constants

## Definitions

Symbol: StieltjesGamma $\gamma_{n}\!\left(a\right)$ Stieltjes constant

## Generating functions

$\zeta\!\left(s\right) = \frac{1}{s - 1} + \sum_{n=0}^{\infty} \frac{{\left(-1\right)}^{n}}{n !} \gamma_{n} {\left(s - 1\right)}^{n}$
$\zeta\!\left(s, a\right) = \frac{1}{s - 1} + \sum_{n=0}^{\infty} \frac{{\left(-1\right)}^{n}}{n !} \gamma_{n}\!\left(a\right) {\left(s - 1\right)}^{n}$

## Limit representations

$\gamma_{n}\!\left(a\right) = \lim_{N \to \infty} \left[\left(\sum_{k=0}^{N} \frac{\log^{n}\!\left(k + a\right)}{k + a}\right) - \frac{\log^{n + 1}\!\left(N + a\right)}{n + 1}\right]$

## Specific values

$\gamma_{n}\!\left(1\right) = \gamma_{n}$
$\gamma_{0}\!\left(1\right) = \gamma_{0} = \gamma$
$\gamma_{0}\!\left(a\right) = -\psi\!\left(a\right)$
$\gamma_{1}\!\left(\frac{1}{2}\right) = \gamma_{1} - 2 \gamma \log(2) - \log^{2}\!\left(2\right)$
Table of $\gamma_{n}$ to 50 digits for $0 \le n \le 30$
Table of $\gamma_{{10}^{n}}$ to 50 digits for $0 \le n \le 20$

## Recurrence relations

$\gamma_{n}\!\left(a + 1\right) = \gamma_{n}\!\left(a\right) - \frac{\log^{n}\!\left(a\right)}{a}$

## Integral representations

$\gamma_{n}\!\left(a\right) = -\frac{\pi}{2 \left(n + 1\right)} \int_{0}^{\infty} \frac{\log^{n + 1}\!\left(a - \frac{1}{2} + i x\right) + \log^{n + 1}\!\left(a - \frac{1}{2} - i x\right)}{\cosh^{2}\!\left(\pi x\right)} \, dx$

## Bounds and inequalities

$\left|\gamma_{n}\right| < {10}^{-4} {e}^{n \log\left(\log(n)\right)}$

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

2020-04-08 16:14:44.404316 UTC