arXiv subjects

Jayashree Kalita

Publications and source records attributed to Jayashree Kalita.

3 recordsLinked to original sources

arXiv2026

Exceptional Sign Pairs in Oscillatory Asymptotics and Conjectures of Andrews

In this paper, we prove an \emph{exceptional sign pair phenomenon} for a general family of integer sequences with exponentially growing oscillatory asymptotics. This resolves, in particular, conjectural sign patterns proposed by Andrews in a 1986 paper for three $q$-series from Ramanujan's Lost Notebook, namely $v_2(q), v_3(q),$ and $v_4(q)$. Recent work of Kundu, Storzer, Wang, and the author established that the coefficients of these $q$-series are alternating in sign except in a density-zero set. We prove here that the exceptional indices where the alternating sign pattern fails occur infinitely often in a structured manner. More precisely, we show that there exist infinitely many pairs of consecutive coefficients having the same sign, and that the magnitude of at least one coefficient in each pair is a local minimum of the sequence of absolute values of the coefficients. Together with earlier results, this completes the resolution of Andrews' conjectures and their analogs for all the $q$-series in his original study.

math.NT↗

arXiv2026

On a conjecture of Andrews and almost alternating sign patterns

In this paper, we prove a sign phenomenon first observed by Andrews for certain $q$-series from Ramanujan's Lost Notebook. For three of the series considered by Andrews, namely $v_2(q)$, $v_3(q)$, and $v_4(q)$, we show that the coefficients are alternating in sign, with only a density-zero set of exceptions. Our approach yields precise asymptotic formulas for the coefficients via an adapted circle method, inspired by the work of Folsom-Males-Rolen-Storzer on the $q$-series $v_1(q)$, revealing an interplay between exponential growth and oscillatory behaviour. This interaction produces a dominant alternating sign factor, which governs the sign regularity observed numerically by Andrews. More broadly, we establish the same sign behaviour for explicit infinite families of $q$-hypergeometric series encompassing these examples, and show that it arises systematically from oscillatory asymptotics of these $q$-series near roots of unity. We introduce an additional family whose coefficients appear to exhibit similar sign regularity, suggesting that this phenomenon is widespread and may point towards a deeper underlying theory.

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arXiv2026

Some convolution identities for mock modular forms arising from the theory of holomorphic projection

Convolution identities and recursive formulas have long played a role in the theory of holomorphic modular forms and their applications. These have served both as striking formulas and as fundamentally useful tools for applications to combinatorics. Recently, there has been renewed interest in examples arising from non-holomorphic modular forms. These include the famous Hurwitz-Kronecker class number relations dating to 1885, and groundbreaking work of Imamoğlu, Raum, and Richter from 2014. Imamoğlu, Raum, and Richter developed a theory of holomorphic projection for products of (vector-valued) harmonic Maass forms and holomorphic modular forms to produce many such formulas. This was related shortly thereafter by Duncan, Griffin, and Ono to replicable-type functions in the sense of Conway and Norton, and such recursions played a key role in their proof of the Umbral Moonshine Conjecture. Here, we develop new results on holomorphic projections of such functions which is more convenient for many natural cases. Imamoğlu, Raum, and Richter's choice corresponds to the case in which the generalized Pell equation $m^2 - Dn^2 = N$ has a square value for $D$, which has only finitely many solutions for a given value of $N$. Our main results cover the cases of general $D$, in particular those for which the Pell equation has infinitely many solutions. As an application, we resolve a recent conjecture of the second author. More generally, our approach yields 18 convolution identities for mock theta functions, which we boil down to 5 identities that can directly be used to prove the others. In the appendices, we also show how these identities can be proven by more direct $q$-series methods; however, the key utility of the holomorphic projection formulas is that they give a tool to automatically discover and verify such formulas.

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