We study the behavior of the signs of the coefficients of certain infinite products involving the Rogers-Ramanujan continued fraction. For example, if $$\sum_{ n=0}^{ \infty}A(n)q^{ n}:= \dfrac{ (q^2;q^5)_\infty^5(q^3;q^5)_\infty^5}{ (q;q^5)_\infty^5(q^4;q^5)_\infty^5},$$then $A(5n+1)>0$, $A(5n+2)>0$, $A(5n+3)>0$, and $A(5n+4)<0$. We also find a few congruences satisfied by some coefficients. For example, for all nonnegative integers $n$, $A(9n+4)\equiv 0 \pmod3$, $ A(16n+13)\equiv 0 \pmod4$, and $A(15n+r)\equiv0\pmod{ 15}$, where $r\in\{ 4, 8, 13, 14\}$.
Categories: math.NT