Generalized Lerch formulas: Examples of zeta-regularized products

We prove∏ˆm=0∞(∏j=1n(m+zj))=∏j=1n2πΓ(zj)=∏j=1n(∏ˆm=0∞(m+zj)), where ∏ˆnan is the zeta-regularized product of the sequence {an}n and Γ(z) is Euler's gamma function. As a part of our result, we obtain the formula of Lerch, Kurokawa and Wakayama. Moreover this result gives an example of a pair of sequences {an},{bn} which satisfies ∏ˆn(an⋅bn)=∏ˆnan⋅∏ˆnbn, although this equality does not hold in general. We also give two-dimensional analogue and q-analogue of our result. Barnes' double gamma functions and Jackson's q-gamma functions appear instead of Euler's gamma function Γ(z).