On the non-quadraticity of values of the q-exponential function and related q-seriesKeijo Väänänen (Oulu) and Wadim Zudilin (Bonn)To our great Peter Bundschuh on his 70th birthday 1. Introduction and main results. Consider the q-exponential function, which is an entire function in the complex z-plane for any q ∈ C, |q| > 1. It is not difficult to adapt the classical proof of the irrationality ofto the case of the number E q (1) for an integer q > 1. Indeed, assuming, by contradiction, that E q (1) = r/s for certain positive integers r and s, we see that the real numberis integral (according to the left-hand side representation) and positive (because of the right-hand side representation), hence it is at least 1, for any 2000 Mathematics Subject Classification: Primary 11J72, 11J82; Secondary 11C20, 15A15, 33D15.
We prove, in a quantitative form, linear independence results for values of a certain class of q -series, which generalize classical q -hypergeometric series. These results refine our recent estimates.
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