Conjectural identities related to the modulo-nine Kanade–Russell identities
Prove the eighteen explicit conjectural sum–product identities arising in the modulo-nine Kanade–Russell program: ten identities obtained from reflections of finite forms under , four identities conjectured by Hickerson in connection with contiguous relations for associated parameterized double sums, and four identities arising from Konenkov’s constructions and their conjugates. The supplied source material does not include the explicit formulas for these eighteen identities.
References
Primary source
Additional references
- Proofs of eighteen conjectural identities related to the modulo nine Kanade–Russell identities — arXiv — Ernest X. W. Xia
Progress summary
A new unrefereed paper claims to prove every identity in this collection, but the claim has not yet been independently checked.
The problem concerns eighteen explicit identities arising from the modulo-nine Kanade–Russell program, including reflected finite forms, identities associated with Hickerson, and constructions of Konenkov. They were longstanding conjectures before the September 2026 preprint by Ernest X. W. Xia.
Known results
- Uncu and Zudilin, 2022: reflection of finite forms produced fifteen conjectural identities, with one proved in each group of three.
- Hickerson-related work, 2022: finite polynomial versions were constructed for four additional conjectural identities, and four reflected identities were proved.
- A separate preprint claims the five original modulo-nine Kanade–Russell identities and related dual identities, but does not cover this full collection.
September 2026 claimed proof
Xia’s preprint Proofs of eighteen conjectural identities related to the modulo nine Kanade–Russell identities claims proofs of all eighteen identities: ten reflected finite-form evaluations, four Hickerson identities, and four identities from Konenkov’s constructions. The work is described as a same-day, unrefereed preprint, with no independent verification or reported error assessment.
Current status (as of September 2026): All eighteen identities are claimed proved in Xia’s unrefereed preprint, but their proofs remain unverified.
Solutions 0
No solutions have been posted yet.