Proof of a conjecture of Baruah and Sarma on sign patterns of certain infinite products

Let $$\sum _{n=0}^{\mathrm{\infty }}A(n)q^n:=\frac{(q^2;q^5{)}_{\mathrm{\infty }}^5(q^3;q^5{)}_{\mathrm{\infty }}^5}{(q;q^5{)}_{\mathrm{\infty }}^5(q^4;q^5{)}_{\mathrm{\infty }}^5},$$ $$\sum _{n=0}^{\mathrm{\infty }}B(n)q^n:=\frac{(q;q^5{)}_{\mathrm{\infty }}^5(q^4;q^5{)}_{\mathrm{\infty }}^5}{(q^2;q^5{)}_{\mathrm{\infty }}^5(q^3;q^5{)}_{\mathrm{\infty }}^5},$$ and $$\sum _{n=0}^{\mathrm{\infty }}D(n)q^n:=\frac{(q^5;q^{25}{)}_{\mathrm{\infty }}(q^{20};q^{25}{)}_{\mathrm{\infty }}}{(q^{10};q^{25}{)}_{\mathrm{\infty }}(q^{15};q^{25}{)}_{\mathrm{\infty }}}\frac{(q^2;q^5{)}_{\mathrm{\infty }}^5(q^3;q^5{)}_{\mathrm{\infty }}^5}{(q;q^5{)}_{\mathrm{\infty }}^5(q^4;q^5{)}_{\mathrm{\infty }}^5}$$ where $(a;q{)}_{\mathrm{\infty }}:={\prod }_{k=0}^{\mathrm{\infty }}(1-a{q}^k)$ and $\mathrm{\mid }q\mathrm{\mid }<1.$ These sequences are closely related to the celebrated Rogers-Ramanujan continued fraction. In this paper, we study the sign behavior o of the coefficients $A(n),B(n)$ and $D(n)\mathrm{.}$ We prove that for all integers $n\ge 0,$ \begin{align*} A(5n)<0\quad(n\neq0),\qquad B(5n) < 0\quad(n\neq0),\qquad D(5n+1)>0. \end{align*} This confirms a recent conjecture of Baruah and Sarma. Our proof is different from the previous method of Baruah and Sarma, and combines asymptotic coefficient analysis with symbolic computation for finite case verification.