Discussion on some conjectures regarding the periodicity of sign patterns of certain infinite products involving the Rogers-Ramanujan Continued Fractions

Abstract

Let $R(q)$ denote the Rogers-Ramanujan continued fraction. Define $$ \frac{1}{R^5(q)}=\displaystyle \sum_{n=0}^{\infty}A(n)q^{n} \quad \text{and} \quad R^5(q)=\displaystyle\sum_{n=0}^{\infty}B(n)q^{n}.$$ Baruah and Sarma recently posed conjectures regarding the sign patterns of $A(5n), B(5n)$ for $n\geq 0.$ In this paper, we show that these conjectures do not hold for $n=0$.