Proofs of Some Kanade--Russell Mod 12 Conjectures
Abstract
Kanade and Russell conjectured seventeen Rogers--Ramanujan type identities of modulus 12. Eleven of these identities involving triple sums were proved by Bringmann--Jennings-Shaffer--Mahlburg and by Rosengren. Motivated by these works and using similar methods, we settle all the six remaining conjectures including two triple sum identities labeled I5a and I6a originated from Russell's thesis and four quadruple sum identities labeled 7, 7a, 8 and 8a. Our proof of the triple sum identities combines linear recurrences, q-difference equations, and q-series summation formulas. For the quadruple sum identities, we represent the sums as contour integrals whose integrands are infinite products and evaluate them by residue calculus. The resulting residues reduce to single sum cubic basic hypergeometric series, and we are able to express them as infinite products.
Related
- Proofs of eighteen conjectural identities related to the modulo nine Kanade--Russell identities
- Proofs of some conjectures by Bringmann, Han, Heim, and Kane on periodic sign changes in eta-quotients
- The three Kanade-Russell identities modulo nine
- Proofs of some OEIS conjectures on Wythoff sums, Fibonacci and Lucas words
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BibTeX
@misc{wang2026proofs,
title = {{Proofs of Some Kanade--Russell Mod 12 Conjectures}},
author = {Liuquan Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07822},
eprint = {2609.07822},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07822},
}
RIS
TY - UNPB
TI - Proofs of Some Kanade--Russell Mod 12 Conjectures
AU - Liuquan Wang
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07822
UR - https://arxiv.org/abs/2609.07822
AB - Kanade and Russell conjectured seventeen Rogers--Ramanujan type identities of modulus 12. Eleven of these identities involving triple sums were proved by Bringmann--Jennings-Shaffer--Mahlburg and by Rosengren. Motivated by these works and using similar methods, we settle all the six remaining conjectures including two triple sum identities labeled I_5a and I_6a originated from Russell's thesis and four quadruple sum identities labeled 7, 7a, 8 and 8a. Our proof of the triple sum identities combines linear recurrences, q-difference equations, and q-series summation formulas. For the quadruple sum identities, we represent the sums as contour integrals whose integrands are infinite products and evaluate them by residue calculus. The resulting residues reduce to single sum cubic basic hypergeometric series, and we are able to express them as infinite products.
ER -