The Corners of Ramanujan: Partial- and False-Theta Companions to Third-Order Mock Theta Functions
Abstract
Most of the traditional third-order mock theta functions occur naturally as specialized values of partition-rank generating functions. The same finite rank kernels also occur in the ordinary and odd unimodal-rank series studied previously in the literature. We organize a parallel collection of corner series by retaining these kernels and replacing their quadratic core weights by linear peak weights. Their root-of-unity specializations are governed by partial and false theta identities recorded in Ramanujan's Lost Notebook; in one exceptional case the corner companion is itself a mock theta function. The classical rank series and Ramanujan identities are used as inputs. Our new results include exact two- and three-color partition identities with sparse theta discrepancies and an oriented three-dimensional corner with a distinct tower. In both color cases, positive decompositions separate the colored companion into copies of the relevant odd-corner series and a corresponding decorated even-corner series. Thus both sparse discrepancies become signed differences between natural corner families. Burson's boxed modular involution provides a common rank-refined model for the two corrections, whose surviving fixed points produce the exceptional quadratic sequences. For the side rank of the three-dimensional corners we obtain an exact formula at z=−1, a strict parity bias, quartic and cubic root-of-unity rank-difference identities, and exact equidistribution modulo $3$ on the progression 6n+5. The cubic identity also gives exact congruence formulas on the progressions 6n+2 and 6n+3. We further classify the associated binary triangular support modulo every integer and derive general congruence families.
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BibTeX
@misc{andrews2026corners,
title = {{The Corners of Ramanujan: Partial- and False-Theta Companions to Third-Order Mock Theta Functions}},
author = {George E. Andrews and Mohamed El Bachraoui},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25380},
eprint = {2610.25380},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25380},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Corners of Ramanujan: Partial- and False-Theta Companions to Third-Order Mock Theta Functions
AU - George E. Andrews
AU - Mohamed El Bachraoui
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.25380
UR - https://arxiv.org/abs/2610.25380
AB - Most of the traditional third-order mock theta functions occur naturally as specialized values of partition-rank generating functions. The same finite rank kernels also occur in the ordinary and odd unimodal-rank series studied previously in the literature. We organize a parallel collection of corner series by retaining these kernels and replacing their quadratic core weights by linear peak weights. Their root-of-unity specializations are governed by partial and false theta identities recorded in Ramanujan's Lost Notebook; in one exceptional case the corner companion is itself a mock theta function. The classical rank series and Ramanujan identities are used as inputs. Our new results include exact two- and three-color partition identities with sparse theta discrepancies and an oriented three-dimensional corner with a distinct tower. In both color cases, positive decompositions separate the colored companion into copies of the relevant odd-corner series and a corresponding decorated even-corner series. Thus both sparse discrepancies become signed differences between natural corner families. Burson's boxed modular involution provides a common rank-refined model for the two corrections, whose surviving fixed points produce the exceptional quadratic sequences. For the side rank of the three-dimensional corners we obtain an exact formula at z=−1, a strict parity bias, quartic and cubic root-of-unity rank-difference identities, and exact equidistribution modulo $3$ on the progression 6n+5. The cubic identity also gives exact congruence formulas on the progressions 6n+2 and 6n+3. We further classify the associated binary triangular support modulo every integer and derive general congruence families.
ER -