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

TL;DR

This work investigates conjectures by Baruah and Sarma on the periodic sign patterns of coefficients in Rogers–Ramanujan–derived infinite products, specifically $1/R^5(q)$ and $R^5(q)$. The authors prove that the proposed sign regularities fail at the initial indices, showing $A(0)>0$, $A(10)<0$, $A(15)<0$, and $B(0)>0$, $B(5)<0$, thereby refuting the conjectures for $A(5n)$ and $B(5n)$ at $n=0$, with $D(1)>0$ also established. They derive exact generating-function expressions for $\sum A(5n)q^n$ and $\sum B(5n)q^n$ by coupling known $q$-product identities with coefficient extraction and linking to the classical sequences $c(n)$ and $d(n)$. The proofs rely on identities such as $\dfrac{1}{R^5(q)}-q^2R^5(q)=11q+\dfrac{f_1^6}{f_5^6}$ and on the quintuple-product framework, clarifying the limitations of observed $5$-step sign periodicity in these Rogers–Ramanujan related coefficients. These results refine our understanding of the arithmetic structure of Rogers–Ramanujan products and their coefficient signs.

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 $n=0$.

Theorems & Definitions (13)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Conjecture 1.5
• Theorem 1.6
• remark 1
• Theorem 1.7
• remark 2
• Theorem 1.8