On the roots of the generalized Rogers-Ramanujan function

Abstract

We give simple proofs of the fact that, for certain parameters, the roots of the generalized Rogers-Ramanujan function are irrational numbers and, for example, that at least one of the following two numbers is irrational: $ \{\sum _{n=1}^\infty {F_n }/({m^n \prod _{i=0}^{n-1}\phi (k+i)}), \sum _{n=1}^\infty {F_n }/(m^n \prod _{i=0}^{n-1}$ $\phi (k+i+1)) \}$, where $F_{n+2}=F_{n+1}+F_n$, $F_0=0,F_1=1$ (the Fibonacci sequence), $m$ is a natural number $> ({1+\sqrt 5})/{2}$ and $\phi (k)$ is any function taking positive integer values such that $\limsup _{k\rightarrow \infty } \phi (k)= \infty $.