Jackson's integral of multiple Hurwitz-Lerch zeta functions and multiple gamma functions

Using the Jackson integral, we obtain the q-integral analogue of Raabe's 1843 formula for Barnes multiple Hurwitz-Lerch zeta functions and Barnes and Vardi's multiple gamma functions. Our results generalize q-integral analogue of the Raabe type formulas for the Hurwitz zeta functions and log gamma functions in [9]. During the proof we also obtain a new formula on the relationship between the higher and lower orders Hurwitz zeta functions.