In mathematics, the Glaisher–Kinkelin constant or Glaisher's constant, typically denoted A, is a mathematical constant, related to special functions like the K-function and the Barnes G-function.
The constant also appears in a number of sums and integrals, especially those involving the gamma function and the Riemann zeta function.
It is named after mathematicians James Whitbread Lee Glaisher and Hermann Kinkelin.
Its approximate value is: Glaisher's constant plays a role both in mathematics and in physics.
It appears when giving a closed form expression for Porter's constant, when estimating the efficiency of the Euclidean algorithm.
It also is connected to solutions of Painlevé differential equations and the Gaudin model.
[1] The Glaisher–Kinkelin constant A can be defined via the following limit:[2] where
An analogous limit, presenting a similarity between
, is given by Stirling's formula as: with
which shows that just as π is obtained from approximation of the factorials, A is obtained from the approximation of the hyperfactorials.
Just as the factorials can be extended to the complex numbers by the gamma function such that
for positive integers n, the hyperfactorials can be extended by the K-function[3] with
also for positive integers n, where: This gives:[1] A related function is the Barnes G-function which is given by and for which a similar limit exists:[2] The Glaisher-Kinkelin constant also appears in the evaluation of the K-function and Barnes-G function at half and quarter integer values such as:[1][4] with
Similar to the gamma function, there exists a multiplication formula for the K-Function.
It involves Glaisher's constant:[5] The logarithm of G(z + 1) has the following asymptotic expansion, as established by Barnes:[6] The Glaisher-Kinkelin constant is related to the derivatives of the Euler-constant function:[5][7]
also is related to the Lerch transcendent:[8] Glaisher's constant may be used to give values of the derivative of the Riemann zeta function as closed form expressions, such as:[2][9] where γ is the Euler–Mascheroni constant.
gives the following series:[2] which directly leads to the following product found by Glaisher: Similarly it is which gives: An alternative product formula, defined over the prime numbers, reads:[10] Another product is given by:[5] A series involving the cosine integral is:[11] Helmut Hasse gave another series representation for the logarithm of Glaisher's constant, following from a series for the Riemann zeta function:[8] The following are some definite integrals involving Glaisher's constant:[1] the latter being a special case of:[12] We further have:[13]
ln π − 1
A double integral is given by:[8] The Glaisher-Kinkelin constant can be viewed as the first constant in a sequence of infinitely many so-called generalized Glaisher constants or Bendersky constants.
[1] They emerge from studying the following product:
Defining the following function
with the Bernoulli numbers
), one may approximate the above products asymptotically via
we get Stirling's approximation without the factor
, similar as in the limit definition of
This leads to the following definition of the generalized Glaisher constants: which may also be written as: This gives
and in general:[1][14][15] with the harmonic numbers
, there exist closed form expressions for
in terms of the values of the Riemann zeta function such as:[1] For odd
in terms of the derivative of the Riemann zeta function such as: The numerical values of the first few generalized Glaisher constants are given below: