Glaisher–Kinkelin Constant
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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Glaisher–Kinkelin constant or Glaisher's constant, typically denoted , is a
mathematical constant A mathematical constant is a number whose value is fixed by an unambiguous definition, often referred to by a special symbol (e.g., an Letter (alphabet), alphabet letter), or by mathematicians' names to facilitate using it across multiple mathem ...
, related to
special functions Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications. The term is defined by ...
like the -function and the Barnes -function. The constant also appears in a number of
sums In mathematics, summation is the addition of a sequence of numbers, called ''addends'' or ''summands''; the result is their ''sum'' or ''total''. Beside numbers, other types of values can be summed as well: functions, vectors, matrices, polynom ...
and
integral In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental oper ...
s, especially those involving the
gamma function In mathematics, the gamma function (represented by Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined ...
and the
Riemann zeta function The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for and its analytic c ...
. It is named after
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
s
James Whitbread Lee Glaisher James Whitbread Lee Glaisher (5 November 1848, in Lewisham — 7 December 1928, in Cambridge) was a prominent English mathematician and astronomer. He is known for Glaisher's theorem, an important result in the field of integer partitions, a ...
and Hermann Kinkelin. Its approximate value is: : = ...   . Glaisher's constant plays a role both in mathematics and in
physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
. It appears when giving a closed form expression for Porter's constant, when estimating the efficiency of the
Euclidean algorithm In mathematics, the Euclidean algorithm,Some widely used textbooks, such as I. N. Herstein's ''Topics in Algebra'' and Serge Lang's ''Algebra'', use the term "Euclidean algorithm" to refer to Euclidean division or Euclid's algorithm, is a ...
. It also is connected to solutions of Painlevé differential equations and the Gaudin model.


Definition

The Glaisher–Kinkelin constant can be defined via the following limit: :A=\lim_ \frac where H(n) is the hyperfactorial: H(n)= \prod_^ i^i = 1^1\cdot 2^2\cdot 3^3 \cdot \cdot n^nAn analogous limit, presenting a similarity between A and \sqrt, is given by Stirling's formula as: :\sqrt=\lim_ \frac with n!= \prod_^ i = 1 \cdot 2\cdot 3\cdot \cdot nwhich shows that just as ''π'' is obtained from approximation of the factorials, ''A'' is obtained from the approximation of the hyperfactorials.


Relation to special functions

Just as the factorials can be extended to the
complex numbers In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
by the
gamma function In mathematics, the gamma function (represented by Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined ...
such that \Gamma(n)=(n-1)! for positive integers ''n'', the hyperfactorials can be extended by the K-function with K(n)= H(n-1) also for positive integers ''n'', where: :K(z)=(2\pi)^ \exp\left binom+\int_0^ \ln \Gamma(t + 1)\,dt\right/math> This gives: :A=\lim_ \frac. A related function is the Barnes -function which is given by :G(n)=\frac and for which a similar limit exists: :\frac1A=\lim_ \frac. The Glaisher-Kinkelin constant also appears in the evaluation of the K-function and Barnes-G function at half and quarter
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
values such as: :K(1/2) = \frac :K(1/4) = A^\exp\left(\frac-\frac\right) :G(1/2) = \frac :G(1/4) = \frac\exp\left(\frac-\frac\right) with G being Catalan's constant and \varpi=\frac being the
lemniscate constant In mathematics, the lemniscate constant is a transcendental mathematical constant that is the ratio of the perimeter of Bernoulli's lemniscate to its diameter, analogous to the definition of for the circle. Equivalently, the perimeter of the ...
. Similar to the
gamma function In mathematics, the gamma function (represented by Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined ...
, there exists a multiplication formula for the K-Function. It involves Glaisher's constant: :\prod_^K\left(\frac jn \right) = A^n^e^ The
logarithm In mathematics, the logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the logarithm of to base is , because is to the rd power: . More generally, if , the ...
of ''G''(''z'' + 1) has the following
asymptotic expansion In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation ...
, as established by Barnes: : \ln G(z+1) = \frac \ln z - \frac + \frac\ln 2\pi -\frac \ln z + \left(\frac-\ln A \right)+\sum_^N \frac+O\left(\frac\right) The Glaisher-Kinkelin constant is related to the derivatives of the Euler-constant function: : \gamma'(-1)= \frac\ln2 + 6\ln A - \frac32 \ln\pi - 1 : \gamma''(-1) = \frac\ln2 + 24\ln A - 4 \ln\pi - \frac- \frac A also is related to the Lerch transcendent: :\frac(-1,-1,1)=3\ln A - \frac13\ln2 - \frac 14 Glaisher's constant may be used to give values of the derivative of the
Riemann zeta function The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for and its analytic c ...
as closed form expressions, such as: :\zeta'(-1)=\frac-\ln A :\zeta'(2)=\frac 6 \left( \gamma+\ln 2\pi - 12 \ln A \right) where is the
Euler–Mascheroni constant Euler's constant (sometimes called the Euler–Mascheroni constant) is a mathematical constant, usually denoted by the lowercase Greek letter gamma (), defined as the limiting difference between the harmonic series and the natural logarith ...
.


Series expressions

The above formula for \zeta'(2) gives the following series: :\sum_^\infty \frac=\frac 6 \left( 12 \ln A - \gamma-\ln 2\pi \right) which directly leads to the following product found by Glaisher: :\prod_^\infty k^\frac = \left(\frac \right)^\frac Similarly it is :\sum_^ \frac=\frac \left( 36 \ln A - 3 \gamma - \ln 16\pi^3 \right) which gives: :\prod_^ k^\frac = \left(\frac \right)^\frac An alternative product formula, defined over the
prime numbers A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
, reads: :\prod_ p^\frac = \frac, Another product is given by: :\prod_^\infty \left(\frac\right)^ = \frac A series involving the cosine integral is: :\sum_^\infty \frac=\frac(4\ln A -1)
Helmut Hasse Helmut Hasse (; 25 August 1898 – 26 December 1979) was a German mathematician working in algebraic number theory, known for fundamental contributions to class field theory, the application of ''p''-adic numbers to local class field theory and ...
gave another series representation for the logarithm of Glaisher's constant, following from a series for the Riemann zeta function: :\ln A=\frac 1 8 - \frac 1 2 \sum_^\infty \frac 1 \sum_^n (-1)^k \binom n k (k+1)^2 \ln(k+1)


Integrals

The following are some definite integrals involving Glaisher's constant: :\int_0^\infty \frac \, dx = \frac 1 -\frac 1 2 \ln A :\int_0^\frac12 \ln\Gamma(x) \, dx = \frac 3 2 \ln A+\frac 5 \ln 2+\frac 1 4 \ln \pi the latter being a special case of: : \int_0^z \ln \Gamma(x)\,dx=\frac+\frac\ln 2\pi +z\ln\Gamma(z) -\ln G(1+z) We further have: \int_0^\infty \frac dx= 3\ln A - \frac 13 \ln2 - \frac 18and \int_0^\infty \frac dx = 3\ln A - \frac\ln2 + \frac 12 \ln \pi -1A double integral is given by: :\int_0^1 \int_0^1 \fracdxdy= 6\ln A - \frac16 \ln 2 - \frac12 \ln\pi - \frac12


Generalizations

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''. They emerge from studying the following product: \prod_^ m^ = 1^ \cdot 2^\cdot 3^\cdot \cdot n^Setting k = 0 gives the factorial n!, while choosing k = 1 gives the hyperfactorial H(n). Defining the following functionP_k(n) = \left(\frac+\frac+\frac\right)\ln n - \frac+k!\sum_^ \frac\frac\left(\ln n + \sum_^j \frac\right)with the
Bernoulli numbers In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, ...
B_k (and using B_1=0), one may approximate the above products asymptotically via \exp(). For k = 0 we get Stirling's approximation without the factor \sqrt as \exp()=n^ e^. For k = 1 we obtain \exp()=n^\,e^, similar as in the limit definition of A. This leads to the following definition of the generalized Glaisher constants: : A_k:=\lim_ \left( e^ \prod_^ m^\right) which may also be written as: : \ln A_k:=\lim_ \left(-P_k(n)+\sum_^ \ln m\right) This gives A_0=\sqrt and A_1=A and in general: : A_k=\exp\left(\fracH_k-\zeta'(-k)\right) with the
harmonic numbers In mathematics, the -th harmonic number is the sum of the Multiplicative inverse, reciprocals of the first natural numbers: H_n= 1+\frac+\frac+\cdots+\frac =\sum_^n \frac. Starting from , the sequence of harmonic numbers begins: 1, \frac, \f ...
H_k and H_0=0. Because of the formula : \zeta'(-2m)=(-1)^m \frac\zeta(2m+1) for m > 0, there exist closed form expressions for A_ with even k=2m in terms of the values of the Riemann zeta function such as: :A_2=\exp\left(\frac\right) :A_4=\exp\left(-\frac\right) For odd k=2m-1 one can express the constants A_ in terms of the derivative of the Riemann zeta function such as: :A_1=\exp\left(-\frac+\frac\right) :A_3=\exp\left(\frac-\frac\right) The numerical values of the first few generalized Glaisher constants are given below:


See also

* Hyperfactorial * Superfactorial *
Stirling's approximation In mathematics, Stirling's approximation (or Stirling's formula) is an asymptotic approximation for factorials. It is a good approximation, leading to accurate results even for small values of n. It is named after James Stirling, though a related ...
*
List of mathematical constants A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. For e ...


References


External links


The Glaisher–Kinkelin constant to 20,000 decimal places
{{DEFAULTSORT:Glaisher-Kinkelin constant Mathematical constants Number theory Glaisher family