The Barnes -function is an analytic continuation of the -function defined in the construction of the Glaisher-Kinkelin constant
(1)
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for , where is the hyperfactorial, which has the special values
(2)
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for integer . This function is a shifted version of the superfactorial (Sloane and Plouffe 1995) with values for , 1, 2, ... given by 0, 1, 1, 1, 2, 12, 288, 34560, 24883200, 125411328000, 5056584744960000, ... (OEIS A000178).
The Barnes -function can arise in spectral functions in mathematical physics (Voros 1987).
It is implemented in the Wolfram Language as BarnesG[n]. A special version of its natural logarithm optimized for large is implemented in the Wolfram Language as LogBarnesG[n].
The Barnes -function for complex may be defined by
(3)
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where is the Euler-Mascheroni constant (Whittaker and Watson 1990, p. 264; Voros 1987). The product can be done in closed form, yielding the identity
(4)
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for , where is the derivative of the Hurwitz zeta function, is the gamma function, and is the Glaisher-Kinkelin constant. Another elegant closed-form expression is given by
(5)
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where is a polygamma function of negative order. The Barnes -function and hyperfactorial satisfy the relation
(6)
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for all complex , where is the log gamma function.
is an entire function analogous to , except that it has order 2 instead of 1.
The Barnes -function is plotted above evaluated at integers values. A slight variant of the integer-valued Barnes -function is sometimes known as the superfactorial.
The Barnes -function satisfies the functional equation
(7)
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and has the Taylor series
(8)
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in . It also gives an analytic solution to the finite product
(9)
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The Barnes -function has the equivalent reflection formulas
(10)
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(11)
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(12)
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(Voros 1987; Whittaker and Watson 1990, p. 264).
The derivative is given by
(13)
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where is the digamma function.
A Stirling-like asymptotic series for as is given by
(14)
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(Voros 1987). This can be made more precise as
(15)
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where is a Bernoulli number (Adamchik 2001b; typo corrected).
has the special values
(16)
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(17)
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(18)
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(OEIS A087013 and A087015) for , where is the gamma function, is Catalan's constant, is the Glaisher-Kinkelin constant, and
(19)
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(20)
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(21)
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(OEIS A087014, A087016, and A087017) for , where is the derivative of the Riemann zeta function evaluated at . In general, for odd ,
(22)
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where
(23)
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for , of which the first few terms are 1, 1, 1, 3, 45, 4725, 4465125, ... (OEIS A057863).
Another G-function is defined by Erdélyi et al. (1981, p. 20) as
(24)
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where is the digamma function. An unrelated pair of functions are denoted and and are known as Ramanujan g- and G-functions.