On irrational valued series involving generalized Fibonacci numbers. II (Q2719876)

By continuing the first part [see ibid. 37, 299-304 (1999; Zbl 0943.11011)], the author gives certain irrationality criteria of infinite series. We quote only the following result: Suppose \(\{U_n\}\) is a generalized Fibonacci sequence generated with respect to the relatively prime pair \((P,Q)\) with \(Q\neq 0\) and \(P>|Q+1|\). Let \(k\in \mathbb{N}\) be given. If \(f:\mathbb{N}\to \mathbb{N}\setminus\{0\}\) has the property \(f(n+1)>2f(n)+2k\) \((n\geq 1)\), then \(\sum^\infty_{n=1} 1/a_n\) is irrational, where \(a_n=U_{f(n)}\dots U_{f(n)+k}\).