Irrationality results for values of generalized Tschakaloff series
The series of the title takes the shape \[ f(x)= \sum^\infty_{n=0} {s_nu_0 u_1\dots u_n\over t_nv_0 v_1\dots v_n} {x^n\over q^{n (n+1)/2}}, \] where \(s_n,t_n, u_n,v_n\) are sequences of non-zero integers restricted in absolute value by \(\exp (cn^\delta)\) with \(0<\delta <{1\over 4}\) and \(c>0\). The main result states that if \(\alpha\) is a non-zero real number and \(q\) is an integer with \(| q|\geq 2\), then \(f(\alpha)\) is irrational and has a measure of irrationality satisfying \[ \left| f(\alpha) -{M\over N}\right |> \exp \bigl\{-c_1 (\log N)^{8/5} \bigr\}, \] where \(c_1\) is a positive constant depending only on \(f\) and \(\alpha\). The result extends and improves a number of results on Tschakaloff's series (the case with \(s_n=t_n =u_n= v_n=1)\) and its extensions. The proof follows closely Mahler's method in transcendental number theory. It exploits the functional equations connecting \(f(qx)\) and \(f(x)\) and is an attractive new application of the method.
scientific article; zbMATH DE number 1309273