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arXiv · 2503.08517

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

Nayandeep Deka Baruah·Abhishek Sarma
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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 $r\in\{4, 8, 13, 14\}$.

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Nayandeep Deka Baruah, Abhishek Sarma. 2025-03-11. Sign Patterns and Congruences of certain infinite products involving the Rogers-Ramanujan continued fraction. https://arxiv.org/abs/2503.08517

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