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

:

OEIS|id=A074962, where ζ denotes the Riemann zeta function and ζ' is its derivative. It also satisfies

:$A\; =\; lim\_\{n\; ightarrowinfty\}frac\{1^1\; 2^2\; 3^3\; cdots\; n^n\}\{n^\{n^2/2+n/2+1/12\}e^\{-n^2/4$

where

:$1^1\; 2^2\; 3^3\; cdots\; n^n=\; K(n+1)$

is the K-function. One also has

:$frac\{e^\{1/12\{A\}\; =\; lim\_\{n\; ightarrowinfty\}frac\{G(n)\}\{n^\{n^2/2-1/12\}(2pi)^\{n/2\}e^\{-3n^2/4$

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

:$A\; =\; 2^\{7/36\}pi^\{-1/6\}expleft\{frac\{1\}\{3\}+frac\{2\}\{3\}int\_0^\{1/2\}\; lnleft\; [Gamma(x+1)\; ight]\; dx\; ight\}.$

A series representation is given by Guillera and 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.

References

* Jesus Guillera and Jonathan Sondow, " [http://arxiv.org/abs/math.NT/0506319 Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent] " "ArXiv" math.NT/0506319 (2005) "(Provides a variety of relationships.)"

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