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

We study the behavior of the signs of the coefficients of certain infinite products involving the Rogers-Ramanujan continued fraction. For example, if ∑ₙ₌₀∞A(n)qⁿ:= (q²;q⁵)_∞⁵(q³;q⁵)_∞⁵/(q;q⁵)_∞⁵(q⁴;q⁵)_∞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)≡ 0 3, A(16n+13)≡ 0 4, and A(15n+r)≡015, where r∈\4, 8, 13, 14\.