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~$\II$, 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.

Keywords:

measure of linear independence, $q$-series measure of linear independence, $q$-series

MSC Classifications:

11J82, 33D15 show english descriptions Measures of irrationality and of transcendence
Basic hypergeometric functions in one variable, ${}_r\phi_s$ 11J82 - Measures of irrationality and of transcendence
33D15 - Basic hypergeometric functions in one variable, ${}_r\phi_s$

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