Proofs of eighteen conjectural identities related to the modulo nine Kanade--Russell identities
Abstract
Uncu and Zudilin studied reflections of finite forms of the modulo nine Kanade--Russell identities under q↦ q−1, leading to ten conjectural sum--product evaluations, including two proposed by Warnaar. Hickerson conjectured four further identities in his study of contiguous relations for the associated parameterized double sums. By constructing finite forms of Hickerson's sums and studying their reflections, Konenkov proposed two additional product identities, whose conjugates give two more. In this paper, we prove all eighteen conjectural identities by combining Nahm-sum dual evaluations established by the author with parameterized divided-difference identities. For the symmetric reflections, Schur-type decompositions and finite rational identities reduce the required evaluations to nonsingular two-by-two linear systems. The asymmetric and cyclotomic cases are treated through q-Airy--theta connection formulas, whereas the Hickerson identities are derived from low-order divided differences.
Related
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BibTeX
@misc{xia2026proofs,
title = {{Proofs of eighteen conjectural identities related to the modulo nine Kanade--Russell identities}},
author = {Ernest X. W. Xia},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24129},
eprint = {2609.24129},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24129},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Proofs of eighteen conjectural identities related to the modulo nine Kanade--Russell identities
AU - Ernest X. W. Xia
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.24129
UR - https://arxiv.org/abs/2609.24129
AB - Uncu and Zudilin studied reflections of finite forms of the modulo nine Kanade--Russell identities under q↦ q^−1, leading to ten conjectural sum--product evaluations, including two proposed by Warnaar. Hickerson conjectured four further identities in his study of contiguous relations for the associated parameterized double sums. By constructing finite forms of Hickerson's sums and studying their reflections, Konenkov proposed two additional product identities, whose conjugates give two more. In this paper, we prove all eighteen conjectural identities by combining Nahm-sum dual evaluations established by the author with parameterized divided-difference identities. For the symmetric reflections, Schur-type decompositions and finite rational identities reduce the required evaluations to nonsingular two-by-two linear systems. The asymmetric and cyclotomic cases are treated through q-Airy--theta connection formulas, whereas the Hickerson identities are derived from low-order divided differences.
ER -