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Barnes G-function

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Plot of the Barnes G function G(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D
Plot of the Barnes G aka double gamma function G(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D
The Barnes G function along part of the real axis

In mathematics, the Barnes G-function 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] It can be written in terms of the double gamma function.

Formally, the Barnes G-function is defined in the following Weierstrass product form:[2]

where is the Euler–Mascheroni constant, exp(x) = ex is the exponential function, and denotes multiplication (capital pi notation).

The integral representation, which may be deduced from the relation to the double gamma function, is

As an entire function, is of order two, and of infinite type. This can be deduced from the asymptotic expansion given below.

Functional equation and integer arguments

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The Barnes G-function satisfies the functional equation

with normalization . Note the similarity between the functional equation of the Barnes G-function and that of the Euler gamma function:

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

(in particular, ) and thus

where denotes the gamma function and denotes the K-function. The functional equation uniquely defines the Barnes G-function if the convexity condition,

is added.[3] Additionally, the Barnes G-function satisfies the duplication formula,[4]

,

where is the Glaisher–Kinkelin constant.

Characterisation

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Similar to the Bohr–Mollerup theorem for the gamma function, for a constant we have for [5]

and for

as .

Reflection formula

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

The proof of this result hinges on the following evaluation of the cotangent integral: introducing the notation for the log-cotangent 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 integral, 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:

Adamchik (2003) has given an equivalent form of the reflection formula, but with a different proof.[6]

Replacing with in the previous reflection formula gives, after some simplification, the equivalent formula shown below

(involving Bernoulli polynomials):

Taylor series expansion

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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

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Like the gamma function, the G-function also has a multiplication formula:[7]

where is a constant given by:

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

Absolute value

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It holds true that , thus . From this relation and by the above presented Weierstrass product form one can show that

This relation is valid for arbitrary , and . If , then the below formula is valid instead:

for arbitrary real y.

Asymptotic expansion

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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 [8] 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 log-gamma integral

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The parametric log-gamma can be evaluated in terms of the Barnes G-function:[9]

A proof of the formula

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 G-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:

And since then,

Taking the logarithm of both sides introduces the analog of the Digamma function ,

where [2][1] [10]

with Taylor series

References

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  1. ^ a b Barnes, E. W. (1900). "The theory of the G-function". Quarterly J. Pure and Appl. Math. 31: 264–314.
  2. ^ a b c Choi, Juensang; Srivastava, H. M. (1999). "Certain classes of series involving the Zeta Function". J. Math. Anal. Applic. 231: 91–117. doi:10.1006/jmaa.1998.6216.
  3. ^ Vignéras, M. F. (1979). "L'équation fonctionelle de la fonction zêta de Selberg du groupe modulaire PSL". Astérisque. 61: 235–249.
  4. ^ Park, Junesang (1996). "A duplication formula for the double gamma function $Gamma_2$". Bulletin of the Korean Mathematical Society. 33 (2): 289–294.
  5. ^ Marichal, Jean Luc; Zenaidi, Naim (2022). A Generalization of Bohr-Mollerup's Theorem for Higher Order Convex Functions (PDF). Developments in Mathematics. Vol. 70. Springer. p. 218. doi:10.1007/978-3-030-95088-0. ISBN 978-3-030-95087-3.
  6. ^ Adamchik, Viktor S. (2003). "Contributions to the Theory of the Barnes function". arXiv:math/0308086.
  7. ^ Vardi, I. (1988). "Determinants of Laplacians and multiple gamma functions". SIAM J. Math. Anal. 19 (2): 493–507. doi:10.1137/0519035.
  8. ^ E. T. Whittaker and G. N. Watson, "A Course of Modern Analysis", CUP.
  9. ^ Neretin, Yury A. (2024). "The double gamma function and Vladimar Alekseevsky". arXiv:2402.07740 [math.HO].
  10. ^ Merkle, Milan; Ribero Merkle, Monica Moulin (2011). "Krull's theory for the double gamma functions". Appl. Math. Comput. 218 (3): 935–943. doi:10.1016/j.amc.2011.01.090. MR 2831334.