Overview
This research investigates the exceptional sign pair phenomenon observed in three specific $q$-series from Ramanujan's Lost Notebook: $v_2(q)$, $v_3(q)$, and $v_4(q)$. The study confirms that the alternating sign pattern of their coefficients fails at particular indices, which Andrews initially noted in 1986. The authors demonstrate that these exceptional indices appear infinitely often in a structured manner, characterized by pairs of consecutive coefficients exhibiting the same sign. Furthermore, at least one coefficient within each such pair is identified as a local minimum when considering the sequence of absolute values of the coefficients.
Research Context
The coefficients of the $q$-series $v_2(q)$, $v_3(q)$, and $v_4(q)$ were previously established to be alternating in sign, except for a density-zero set of indices. This prior work by Kundu, Storzer, Wang, and the current author provided the foundation upon which this study's investigation into the ‘exceptional indices’ is built. The specific behavior at these exceptional points, where the alternating sign pattern breaks, is the central focus. Andrews first observed this phenomenon in a 1986 paper, highlighting its long-standing presence in the study of these $q$-series.
Approach
The research methodology combines precise asymptotic expansions for the coefficients of $v_2(q), v_3(q)$, and $v_4(q)$ with a detailed analysis of an oscillatory factor. This oscillatory factor, embedded within the expansions, is explicitly linked to governing the exceptional sign behavior observed in the coefficients. By carefully analyzing this factor, the authors were able to characterize the nature and frequency of the deviations from the typical alternating sign pattern.
Findings
- The study proves the existence of an exceptional sign pair phenomenon for $v_2(q)$, $v_3(q)$, and $v_4(q)$. This phenomenon refers to instances where the alternating sign pattern of the coefficients fails.
- These exceptional indices, where the sign pattern deviates, occur infinitely often.
- The occurrence of these exceptional indices is structured.
- Specifically, there are infinitely many pairs of consecutive coefficients that share the same sign.
- Within each of these pairs of same-sign consecutive coefficients, at least one coefficient represents a local minimum in the sequence of absolute values of the coefficients.
- The proof relied on precise asymptotic expansions for the coefficients.
- A careful analysis of an oscillatory factor within these expansions was crucial, as this factor governs the exceptional sign behavior.