# Duplicating Barnes

I have mentioned the Barnes function already once recently, as it arose in a marvelous identity of Widom. Today, here is a look at the duplication formula…

Recall that the Barnes function is defined by

$G(z+1)=(2\pi)^{z/2}e^{-(z(z+1)+\gamma z^2)/2}\prod_{n\geq 1}{(1+z/n)^n\exp(-z+z^2/(2n)}}$

for any complex number z (it is indeed an entire function; here γ is the Euler constant). Besides the ratio

$\frac{G(1+z)^2}{G(1+2z)}$

which appeared in the previous post, the following expression

$f_{Sp}(z)=2^{z^2/2}\frac{G(1+z)\sqrt{\Gamma(1+z)}}{\sqrt{G(1+2z)\Gamma(1+2z)}}$

also occurs naturally in the asymptotic formula of Keating-Snaith for characteristic polynomials of unitary symplectic matrices. For some reasons, this didn’t look nice enough to me (e.g., the computation of

$f_{Sp}(n)=\frac{1}{(2n-1)(2n-3)^2\cdots 1^n}$

for n a positive integer is somewhat laborious). But because of the shape of the term in the square root of the denominator, one can try to simplify this using the basic relation

$G(1+2z)\Gamma(1+2z)=G(2+2z)=G(2(1+z)),$

and the duplication formula for G(2w), which “must exist”, since the Barnes function generalizes the Gamma function, for which it is well-known that

$\Gamma(2z)=\pi^{-1/2}2^{2z-1}\Gamma(z)\Gamma(z+1/2)$

(usually attributed to Legendre).

So there is indeed a duplication formula; I picked it up from this paper of Adamchik (where it is cleaner than in the paper of Vardi referenced by Wikipedia), who states it as follows:

$G(2w)=e^{-3\zeta^{\prime}(-1)}2^{2w^2-2w+5/12}(2\pi)^{(1-2w)/2}G(w)G(w+1/2)^2G(w+1).$

This is promising since in the case considered, the terms involving G, after application of the formula to w=1+z, become

$G(z+3/2)^2G(z+1)G(z+2)=\Gamma(z+1)G(z+1)^2G(z+3/2)^2,$

leading to a cancellation with both the gamma and G-factors in the numerator!

And although the term ζ'(-1) is maybe a bit surprising, a moment’s thought shows it can be eliminated by simply plugging in any specific value of z and using the resulting formula to express it in terms of a specific value of G(z). Indeed, taking z=1/2, and watching the dust settling lazily across the page, we get the very nice expression

$f_{Sp}(z)=2^{-z^2/2-z-1/2}(2\pi)^{(z+1)/2}\frac{G(1/2)}{G(z+3/2)}.$

In fact, it’s clear then that (at least for such purposes), it is best to write the duplication formula in a way which avoids ζ'(-1) altogether (losing a bit of information, since it’s somewhat interesting to know that this quantity is linked to G(1/2)):

$G(1/2)^2G(2w)=(2\pi)^{-w}2^{2w^2-2w+1}\Gamma(w)((G(w)G(w+1/2))^2.$

(This is still not tautological for z=1/2: it contains the value Γ(1/2)=π1/2.) From the shape of this, number theorists, at least, would probably be curious to see what happens if one replaces the Gamma function with

$\Gamma_{\mathbf{C}}(w)=(2\pi)^{-w}\Gamma(w)$

which is the factor at infinity for the local field C. In that case, it seems natural to introduce

$G_{\mathbf{C}}(w)=(2\pi)^{-z(z-1)/2}G(w),$

for which we retain the induction relation of G with respect to Γ:

$G_{\mathbf{C}}(w+1)=\Gamma_{\mathbf{C}}(w)G_{\mathbf{C}}(w).$

In terms of these functions, the duplication formula is even nicer:

$G_{\mathbf{C}}(1/2)^2G_{\mathbf{C}}(2w)=2^{2w^2-2w+1}\Gamma_{\mathbf{C}}(w)((G_{\mathbf{C}}(w)G_{\mathbf{C}}(w+1/2))^2.$