Integral representations of $q$-analogues of the Barnes multiple zeta functions

Abstract

Integral representations of $q$-analogues of the Barnes multiple zeta functions are studied. The integral representation provides a meromorphic continuation of the $q$-analogue to the whole plane and describes its poles and special values at non-positive integers. Moreover, for any weight, employing the integral representation, we show that the $q$-analogue converges to the Barnes multiple zeta function when $q \uparrow 1$ for all complex numbers.