Barnes G-function

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In mathematics, the Barnes G-function G(z) is a function that is an extension of superfactorials to the complex numbers. It is related to the Gamma function, the K-function and the Glaisher–Kinkelin constant, and was named after mathematician Ernest William Barnes.[1] Up to elementary factors, it is a special case of the double gamma function.

Formally, the Barnes G-function is defined in the following Weierstrass product form:

where is the Euler–Mascheroni constant, exp(x) = ex, and ∏ is capital pi notation.

Functional equation and integer arguments[edit]

The Barnes G-function satisfies the functional equation

with normalisation G(1) = 1. Note the similarity between the functional equation of the Barnes G-function and that of the Euler Gamma function:

The functional equation implies that G takes the following values at integer arguments:

(in particular, ) and thus

where denotes the Gamma function and K denotes the K-function. The functional equation uniquely defines the G function if the convexity condition: is added.[2]

Reflection formula 1.0[edit]

The difference equation for the G function, in conjunction with the functional equation for the Gamma function, can be used to obtain the following reflection formula for the Barnes G function (originally proved by Hermann Kinkelin):

The logtangent integral on the right-hand side can be evaluated in terms of the Clausen function (of order 2), as is shown below:

The proof of this result hinges on the following evaluation of the cotangent integral: introducing the notation for the logtangent integral, and using the fact that , an integration by parts gives

Performing the integral substitution gives

The Clausen function – of second order – has the integral representation

However, within the interval , the absolute value sign within the integrand can be omitted, since within the range the 'half-sine' function in the integral is strictly positive, and strictly non-zero. Comparing this definition with the result above for the logtangent itegral, the following relation clearly holds:

Thus, after a slight rearrangement of terms, the proof is complete:

Using the relation and dividing the reflection formula by a factor of gives the equivalent form:


Ref: see Adamchik below for an equivalent form of the reflection formula, but with a different proof.

Reflection formula 2.0[edit]

Replacing z with (1/2) − z'' in the previous reflection formula gives, after some simplification, the equivalent formula shown below (involving Bernoulli polynomials):

Taylor series expansion[edit]

By Taylor's theorem, and considering the logarithmic derivatives of the Barnes function, the following series expansion can be obtained:

It is valid for . Here, is the Riemann Zeta function:

Exponentiating both sides of the Taylor expansion gives:

Comparing this with the Weierstrass product form of the Barnes function gives the following relation:

Multiplication formula[edit]

Like the Gamma function, the G-function also has a multiplication formula:[3]

where is a constant given by:

Here is the derivative of the Riemann zeta function and is the Glaisher–Kinkelin constant.

Asymptotic expansion[edit]

The logarithm of G(z + 1) has the following asymptotic expansion, as established by Barnes:

Here the are the Bernoulli numbers and is the Glaisher–Kinkelin constant. (Note that somewhat confusingly at the time of Barnes [4] the Bernoulli number would have been written as , but this convention is no longer current.) This expansion is valid for in any sector not containing the negative real axis with large.

Relation to the Loggamma integral[edit]

The parametric Loggamma can be evaluated in terms of the Barnes G-function (Ref: this result is found in Adamchik below, but stated without proof):

The proof is somewhat indirect, and involves first considering the logarithmic difference of the Gamma function and Barnes G-function:

where

and is the Euler–Mascheroni constant.

Taking the logarithm of the Weierstrass product forms of the Barnes function and Gamma function gives:

A little simplification and re-ordering of terms gives the series expansion:

Finally, take the logarithm of the Weierstrass product form of the Gamma function, and integrate over the interval to obtain:

Equating the two evaluations completes the proof:

References[edit]

  1. ^ E. W. Barnes, "The theory of the G-function", Quarterly Journ. Pure and Appl. Math. 31 (1900), 264–314.
  2. ^ M. F. Vignéras, L'équation fonctionelle de la fonction zêta de Selberg du groupe mudulaire SL, Astérisque 61, 235–249 (1979).
  3. ^ I. Vardi, Determinants of Laplacians and multiple gamma functions, SIAM J. Math. Anal. 19, 493–507 (1988).
  4. ^ E. T. Whittaker and G.N.Watson, "A course of modern analysis", CUP.