On q-basic multiple gamma functions

We define a q-analogue of the basic multiple gamma function double struck G sing_m(x) introduced in [17] which differs from the one defined by Barnes via the zeta regularized product. We call it a q-basic multiple gamma function. Using this q-basic multiple Gamma function we introduce a q-analogue of the multiple sine function of order m+1. We study properties of such functions from the periodicity, differential-difference equations and duplication formulas, etc. points of view.