Generalized Lerch formulas: Examples of zeta-regularized products

Abstract

We prove ∏ ˆ m = 0 ∞ ( ∏ j = 1 n ( m + z j ) ) = ∏ j = 1 n 2 π Γ ( z j ) = ∏ j = 1 n ( ∏ ˆ m = 0 ∞ ( m + z j ) ) , where ∏ ˆ n a n is the zeta-regularized product of the sequence { a n } 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 { a n } , { b n } which satisfies ∏ ˆ n ( a n ⋅ b n ) = ∏ ˆ n a n ⋅ ∏ ˆ n b n , 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 ) .