Glaisher-Kinkelin constant

In mathematics, the Glaisher-Kinkelin constant, typically denoted A, is a mathematical constant, related to the K-function and the Barnes G-function. It is named after mathematicians James Whitbread Lee Glaisher and Hermann Kinkelin.

The constant can be defined as

$\begin{matrix} A&=&\exp\left(1/12-\zeta^\prime(-1)\right) \\ &=&1.28242712... \end{matrix}$

(sequence A074962 in OEIS), where ζ denotes the Riemann zeta function and ζ' is its derivative. It also satisfies

$A = \lim_{n\rightarrow\infty} \frac{1^1 2^2 3^3 \ldots n^n} {n^{n^2/2+n/2+1/12}e^{-n^2/4}}$

where

$1^1 2^2 3^3 \ldots n^n= K(n+1)$

is the K-function. One also has

$\frac{e^{1/12}}{A} = \lim_{n\rightarrow\infty} \frac{G(n)}{n^{n^2/2-1/12}(2\pi)^{n/2}e^{-3n^2/4}}$

where G is the Barnes G-function. One also has

$A = 2^{7/36}\pi^{-1/6}\exp\left\{\frac{1}{3}+\frac{2}{3}\int_0^{1/2} \ln\left[\Gamma(x+1)\right]dx\right\}.$

A series representation is given by Sondow:

$\ln A - \frac{1}{8} = \frac{1}{2} \sum_{n=0}^\infty \frac{1}{n+1} \sum_{k=0}^n (-1)^{k+1} {n \choose k} (k+1)^2 \ln(k+1)$

The constant also appears in a number of other sums and integrals, especially those involving Gamma functions and zeta functions.

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