Shintani-Barnes zeta and gamma functions

We show that Shintani's work on multiple zeta and gamma functions can be simplified and extended by exploiting difference equations. We re-prove many of Shintani's formulas and prove several new ones. Among the latter is a generalization to the Shintani-Barnes gamma functions of Raabe's 1843 formula integral(0)(1) log Gamma(x) dx = log root2pi, and a further generalization to the Shintani zeta functions. These explicit formulas can be interpreted as "vanishing period integral" side conditions for the ladder of difference equations obeyed by the multiple gamma and zeta functions. We also relate Barnes' triple gamma function to the elliptic gamma function appearing in connection with certain integrable systems. (C) 2003 Elsevier Inc. All rights reserved.