On the roots of the 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: {∑∞n=1Fn/(mn∏n−1i=0ϕ(k+i)),∑∞n=1Fn/(mn∏n−1i=0 ϕ(k+i+1))}, where Fn+2=Fn+1+Fn, F0=0,F1=1 (the Fibonacci sequence), m is a natural number >(1+5–√)/2 and ϕ(k) is any function taking positive integer values such that lim supk→∞ϕ(k)=∞.