Proof of a conjecture of Baruah and Sarma on sign patterns of certain infinite products
Let \[ \sum_{n=0}^{\infty}A(n)q^{n} := \frac{(q^{2};q^{5})_{\infty}^{5}(q^{3};q^{5})_{\infty}^{5}}{(q;q^{5})_{\infty}^{5}(q^{4};q^{5})_{\infty}^{5}}, \] \[ \sum_{n=0}^{\infty} B(n)q^{n} := \frac{(q;q^{5})_{\infty}^{5} (q^{4};q^{5})_{\infty}^{5}} {(q^{2};q^{5})_{\infty}^{5}(q^{3}; q^{5})_{\infty}^{5}}, \] and \[ \sum_{n=0}^{\infty} D(n)q^{n} := \frac{(q^{5};q^{25})_{\infty}(q^{20}; q^{25})_{\infty}} {(q^{10};q^{25})_{\infty}(q^{15}; q^{25})_{\infty}} \frac{(q^{2}; q^{5})_{\infty}^{5}(q^{3};q^{5})_{\infty}^{5}} {(q;q^{5})_{\infty}^{5} (q^{4};q^{5})_{\infty}^{5}} \] where $(a;q)_{\infty} := \prod_{k=0}^{\infty}(1-aq^{k})$ and $|q|<1.$ 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 $D(n).$ We prove that for all integers $n\geq0,$ \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.