Huang-Jiang-Oblomkov conjecture for a = 3: a new infinite family of Rogers-Ramanujan identities, with a Lean certificate conditional on eight literature facts

Kenny Lau and Ken Ono prove the $a = 3$ layer of the Huang-Jiang-Oblomkov conjecture, which equates a count of commuting nilpotent matrix pairs with $X^3 = Y^b$ over finite fields with an explicit theta quotient, giving a new infinite family of Rogers-Ramanujan identities; AxiomProver produced a Lean certificate of the deduction from eight cited literature facts.

What was found

For coprime $1 < a < b$, Huang, Jiang and Oblomkov assembled the normalized counts of commuting nilpotent $n \times n$ matrix pairs over $\mathbb{F}_q$ with $X^a = Y^b$ into a $q$-series $Z_{a,b}(q)$ and conjectured that it equals an explicit product $P_{a,b}(q)$ built from Jacobi's theta function and Dedekind's eta function, which would make the point count essentially a modular function on $\Gamma(a+b)$. The conjecture is layered in $a$: the $a = 2$ layer is classical, containing the Rogers-Ramanujan and Andrews-Gordon identities, and the abstract says nothing was known for $a \ge 3$. This paper proves the $a = 3$ layer in full, which the abstract describes as a new infinite family of Rogers-Ramanujan identities and a geometric origin for Warnaar's products. A companion paper by Huang, Lau, Ono and Paule (arXiv:2608.15219) proves the cases $(a, b) = (3, 4)$, $(3, 5)$, $(3, 7)$ and $(3, 8)$ through a finer sum-to-sum identity, and 2026-09-17-rogers-ramanujan-torus-knot later claims every coprime pair, which contains this layer.

Novelty check

Read the arXiv abstracts of arXiv:2608.05480 and of the companion arXiv:2608.15219, and the Lean repository's formalization.yaml, on 2026-10-16. The first states that nothing was known for $a \ge 3$, and the companion's abstract agrees that no case with $a > 2$ was known. The registry's 2026-09-17-rogers-ramanujan-torus-knot, which claims every coprime pair, cites this paper as a predecessor; no earlier claim of the $a = 3$ layer was found in the registry. This was not a systematic review of the Rogers-Ramanujan literature, and the year the conjecture was posed is left unrecorded, as on the general entry.

Also recorded at

Graded claimed for verification and ai-assisted for autonomy. What these mean.

Cite this entry

whataifound.org. (2026). Huang-Jiang-Oblomkov conjecture for a = 3: a new infinite family of Rogers-Ramanujan identities, with a Lean certificate conditional on eight literature facts. whataifound.org: A Registry of AI Scientific and Mathematical Discoveries. https://whataifound.org/finding/2026-08-06-hjo-a3-rogers-ramanujan

@misc{whataifound-axiommath-2026-ramanujan,
  title        = {Huang-Jiang-Oblomkov conjecture for a = 3: a new infinite family of Rogers-Ramanujan identities, with a Lean certificate conditional on eight literature facts},
  author       = {{whataifound.org}},
  year         = {2026},
  howpublished = {whataifound.org: A Registry of AI Scientific and Mathematical Discoveries},
  note         = {Result by Axiom Math. Verification: Claimed. Autonomy: AI-assisted.},
  url          = {https://whataifound.org/finding/2026-08-06-hjo-a3-rogers-ramanujan}
}

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