Barnes again – E. Kowalski's blog

Continuing our popular series of posts on the Barnes function (here and there), here is a useful remark which I picked up in a 1996 paper of Ehrhardt and Silbermann, Toeplitz determinants with one Fisher-Hartwig singularity: instead of the Weierstrass product expansion

$G(z+1)=(2\pi)^{z/2}e^{-\frac{1}{2}(z(z+1)+\gamma z^2)}\prod_{k\geq 1}{\Bigl(1+\frac{z}{k}\Bigr)^ke^{-z+\frac{z^2}{2k}}},$

where γ is the Euler constant, it may be better to use the alternate product expansion

$G(z+1)=(2\pi)^{z/2}e^{-z(z+1)/2}\prod_{k\geq 1}{\Bigl(1+\frac{z}{k}\Bigr)^k\Bigl(1+\frac{1}{k}\Bigr)^{z^2/2}e^{-z}}$

where the Euler constant is not present anymore. This, in fact, can probably be considered to be the “right” analogue of Euler’s original definition of the Gamma function

$\frac{1}{\Gamma(z+1)}=\prod_{k\geq 1}{\Bigl(1+\frac{z}{k}\Bigr)\Bigl(1+\frac{1}{k}\Bigr)^{-z}$

(which I’ve also discussed earlier, as it occurred in a computation where it was much more to the point than the Weierstrass product).

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