Irrationality of certain fast converging series and infinite products (Q6566518)

For every rational number \(r\) denote \(\| r\| =\max (d,| r|)\) where \(d\) be the least positive integer such that \(dr\) is integer. Let \((u_n)\) be a sequence of positive integers and \((r_n)\) be a sequence of rational numbers. Let \(\alpha\) be a positive rational number and \(\gamma\) be a real number such that \(0\leq \gamma<2\). Assume that \(\lim_{n\to\infty}u_n=\infty\), \(u_{n+1}=\alpha u_n^2+O(u_n^\gamma)\) and \(\log (\| r\| )=o(2^n)\). Then the author proves that \(\sum_{n=0}^\infty \frac {r_n}{u_n}\) is rational number iff either \(r_n=0\) for all \(n>n_0\) or \(u_{n+1}=\alpha u_n^2+\frac {r_{n+1}}{r_n}u_n+\frac {r_{n+2}}{\alpha r_{n+1}}\) for all \(n>n_0\). In addition suppose that for every non-negative integer \(n\) we have \(r_n\not= -u_n\). Then the author proves that the number \(\prod_{n=0}^\infty (1+\frac {r_n}{u_n})\) is rational if and only if either \(r_n=0\) for all \(n>n_0\) or \(u_{n+1}=\alpha u_n^2-\alpha(\frac {r_{n+1}}{\alpha r_n}-r_n)u_n+\frac {r_{n+2}}{\alpha r_{n+1}}-r_{n+1}\) for all \(n>n_0\).\N\NHe also proves that the irrationality measure exponent is less than or equal to \(\frac 8{\min (1,2-\gamma)}+1\) in the case that product or sum is irrational.