# alternating sum with Barnes G functions

MathOverflow Asked by JM Landsberg on September 21, 2020

Let $$G(n)=(n-2)!(n-3)!cdots 1!$$ denote the Barnes G-function.
I am pretty sure that
$$sum_{m=0}^{k^2-1} (-1)^mbinom{k^2-1}m frac{G(k+n-m+1)}{G(n-m+1)G(k+1)(k^2)!} = n-2k^2-2k$$ when $$k$$ is odd
and is
$$n-frac 12(k^2-1)$$ when $$k$$ is even,
but I lack a proof. I’d already be happy for some references for identities regarding the Barnes function, as this is all new to me.

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