Bailey hierarchies for the Andrews–Gordon companion conjecture

Prove the Bailey-generated polynomial identity in Conjecture (conj1_Bailey) for all n≥0, p≥1, k≥4, and 1≤i<k.

Background

Applying the Berkovich–Uncu Bailey transformation to the conjectured finite Andrews–Gordon companion identity produces a further hierarchy involving an outer chain of variables m_p≥⋯≥m_1≥0.

The paper verifies this Bailey-generated identity for k=2, 3, and 4. The conjectural extension to k≥4 would establish infinitely many related polynomial identities once the underlying companion hierarchy is proved.

References

Now we apply Theorem~\ref{thm:BU} with $a=0$ to Conjecture~\ref{conj1} to see the whole infinite hierarchy of polynomial identities seeds from it.

— A MacMahon Analysis View of Cylindric Partitions  (2501.19272 - Li et al., 31 Jan 2025) in Section 7, New Infinite Hierarchies Using Bailey Machinery, Conjecture (conj1_Bailey), equation (LiUncu_FinAndrewsGordon_Bailey)