A Multiple q-translation Formula and Its Implications

Using Hartogs’ fundamental theorem for analytic functions in several complex variables, we establish a multiple q-exponential differential operational identity for the analytic functions in several variables, which can be regarded as a multiple q-translation formula. This multiple q-translation formula is a fundamental result and play a pivotal role in q-mathematics. Using this q-translation formula, we can easily recover many classical conclusions in q-mathematics and derive some new q-formulas. Our work reveals some profound connections between the theory of complex functions in several variables and q-mathematics.