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This is a Quantitative Study study.

September 29, 2025Open Access

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

Key Points

Positive coefficients are found for A(5n+1), A(5n+2), and A(5n+3), while A(5n+4) is negative.
Congruences show A(9n+4) is zero modulo 3, A(16n+13) is zero modulo 4, and A(15n+r) is zero modulo 15.
The study explores the behavior of coefficients in the context of the Rogers-Ramanujan continued fraction.
Analysis of these properties can lead to deeper insights in number theory and related areas.

Abstract

We study the behavior of the signs of the coefficients of certain infinite products involving the Rogers-Ramanujan continued fraction. For example, if ₍=₀^A (n) q^n: = (q²;q⁵) _⁵ (q³;q⁵) _⁵ (q;q⁵) _⁵ (q⁴;q⁵) _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) 0 3, A (16n+13) 0 4, and A (15n+r) 015, where r\4, 8, 13, 14\.

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Cite This Study

Baruah et al. (Tue,) studied this question.

synapsesocial.com/papers/68da58c9c1728099cfd10adf https://doi.org/https://doi.org/10.48550/arxiv.2503.08517

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