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.

Published by

Kowalski

I am a professor of mathematics at ETH Zürich since 2008.