Baker-type estimates for linear forms in the values of q-series

We obtain lower estimates for the absolute values of linear forms of the values of generalized Heine series at non-zero points of an imaginary quadratic field I, in particular of the values of q-exponential function. These estimates depend on the individual coefficients, not only on the maximum of their absolute values. The proof uses a variant of classical Siegel’s method applied to a system of functional Poincar´e-type equations and the connection between the solutions of these functional equations and the generalized Heine series