Sign Patterns and Congruences of certain infinite products involving the Rogers-Ramanujan continued fraction

TL;DR

The paper analyzes sign patterns and congruences of coefficients arising from infinite products tied to the Rogers–Ramanujan continued fraction, focusing on sequences $A(n)$, $B(n)$, $C(n)$, and $D(n)$ defined through $1/R^5(q)$, $R^5(q)$, $R^5(q)/R(q^5)$, and $R(q^5)/R^5(q)$. Employing Ramanujan theta functions, 2-, 3-, and 5-dissection techniques, and intricate $q$-product manipulations, the authors derive a consistent 5-term periodic structure for the signs of these sequences and establish a suite of congruences modulo $3$, $4$, $15$, and related moduli. They prove precise relations among the sequences (e.g., $B(5n+1)=-A(5n+3)$ and $D(5n)=C(5n)$) and demonstrate that many subsequences vanish modulo small primes in a periodic fashion. The results deepen the understanding of sign behavior in Rogers–Ramanujan–themed products and hint at broader 5-periodicity and further modular constraints, while posing conjectures to guide future work.

Abstract

We study the behavior of the signs of the coefficients of certain infinite products involving the Rogers-Ramanujan continued fraction. For example, if $$\sum_{n=0}^{\infty}A(n)q^{n}:= \dfrac{(q^2;q^5)_\infty^5(q^3;q^5)_\infty^5}{(q;q^5)_\infty^5(q^4;q^5)_\infty^5},$$then $A(5n+1)>0$, $A(5n+2)>0$, $A(5n+3)>0$, and $A(5n+4)<0$. We also find a few congruences satisfied by some coefficients. For example, for all nonnegative integers $n$, $A(9n+4)\equiv 0 \pmod3$, $ A(16n+13)\equiv 0 \pmod4$, and $A(15n+r)\equiv0\pmod{15}$, where $r\in\{4, 8, 13, 14\}$.

Theorems & Definitions (15)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• remark 1
• Theorem 1.5
• Theorem 1.6
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3