PreprintCC BY 4.0arXiv
Solutions for Hecke Sum Questions of Banerjee and Bringmann
Abstract
We previously conjectured a Hecke-type identity for a series arising from two-color partitions. Banerjee and Bringmann proved the conjecture using modular methods and asked for a direct proof using Bailey pairs. They also suggested investigating the odd part of the series. We answer both questions by applying Bailey's transform to a Bailey pair of Andrews and a conjugate Bailey pair of Lovejoy. Our main theorem gives formulas for the even and odd parts of a series Pa(q) depending on a parameter a. The specialization a=1 gives the conjectured formula and a companion formula for the odd part.
Related
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BibTeX
@misc{andrews2026solutions,
title = {{Solutions for Hecke Sum Questions of Banerjee and Bringmann}},
author = {George E. Andrews and Mohamed El Bachraoui},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.15107},
eprint = {2605.15107},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.15107},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Solutions for Hecke Sum Questions of Banerjee and Bringmann
AU - George E. Andrews
AU - Mohamed El Bachraoui
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2605.15107
UR - https://arxiv.org/abs/2605.15107
AB - We previously conjectured a Hecke-type identity for a series arising from two-color partitions. Banerjee and Bringmann proved the conjecture using modular methods and asked for a direct proof using Bailey pairs. They also suggested investigating the odd part of the series. We answer both questions by applying Bailey's transform to a Bailey pair of Andrews and a conjugate Bailey pair of Lovejoy. Our main theorem gives formulas for the even and odd parts of a series P_a(q) depending on a parameter a. The specialization a=1 gives the conjectured formula and a companion formula for the odd part.
ER -