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 $$\frac{1}{R^5(q)}=\sum _{n=0}^{\infty }A(n)q^n\text{and}{R}^5(q)=\sum _{n=0}^{\infty }B(n)q^n.$$ Baruah and Sarma recently posed conjectures regarding the sign patterns of $A(5n),B(5n)$ for $n\ge 0.$ In this paper, we show that these conjectures do not hold for $n=0$.