Sign Patterns and Congruences of certain infinite products involving the Rogers-Ramanujan continued fraction

Abstract

We study the behavior of the signs of the coefficients of certain infinite products involving the Rogers-Ramanujan continued fraction. For example, if $$\sum _{n=0}^{\infty }A(n)q^n:=\frac{(q^2;q^5{)}_{\infty }^5(q^3;q^5{)}_{\infty }^5}{(q;q^5{)}_{\infty }^5(q^4;q^5{)}_{\infty }^5},$$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 $n$, $A(9n+4)\equiv 0(\mathrm{mod}3)$, $A(16n+13)\equiv 0(\mathrm{mod}4)$, and $A(15n+r)\equiv 0(\mathrm{mod}15)$, where $r\in \{4,8,13,14\}$.

To appear in The Ramanujan Journal