On the non-quadraticity of values of the q-exponential function and related q-series

Abstract

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.

On the irrationality of generalized q-logarithm

“…In this note we combine the Padé approximation construction from [13] with a version of Bézivin's method as developed in [23] to prove the following general result.…”

“…where the derivative is with respect to z. Though the method from [3,4] underwent modifications and generalizations in the later works of Bézivin himself and other authors [5,6,14,23,30,31], it essentially serves the original class of functions.…”

Irrationality Measures of Numbers Related to Some $q$-Basic Hypergeometric Series

“…We note that there are a lot of works considering arithmetic properties of different type of q-series, see e.g. [14] for a survey of such results, and [3], [4], [5], [6], [8], [10], [11], [12] and [15] for some more recent results, but only a few study functions (2), see [7] and [10]. In [7] Chirskii considered the case K = Q with finite p and proved the following result.…”

Arithmetic properties of cubic and biquadratic theta series

“…Bézivin [3] proved the nonquadraticity of values of the more general Tschakaloff function T q (z) = ∞ n=0 z n q −n(n−1)/2 . The results of Bézivin have been simplified by Bradshaw [4,Chapter 3] and extended by some authors [12]. Last but not least, a celebrated result of Nesterenko [13] implies that θ 2 (q) is transcendental for all nonzero algebraic q satisfying |q| > 1 [2, Theorem 4].…”