Bailey hierarchies for the Bressoud companion conjecture

Prove the Bailey-generated polynomial identity in Conjecture (conj1_Bailey) for all n≥0, p≥1, k≥4, and 1≤i<k, with the modified terminal q^2-binomial factor and the stated alternating bilateral sum.

Background

The paper applies Bailey machinery to the conjectured finite Bressoud companion identity, producing a second family of polynomial identities with an outer multiple sum indexed by p.

The identity is proved only for k=2, 3, and 4 in the paper. Its extension to k≥4 remains conjectural and would supply a Bailey-lemma hierarchy parallel to the Andrews–Gordon companion hierarchy.

References

Theorem~\ref{thm:FodaQuano2} allows us to employ $a=1$ cases of Theorem~\ref{thm:BU}. However, the equation eq:FodaQuano_FinBressoud is limited in other ways.

— A MacMahon Analysis View of Cylindric Partitions  (2501.19272 - Li et al., 31 Jan 2025) in Section 7, New Infinite Hierarchies Using Bailey Machinery, final conjecture before the concluding theorem, equation (LiUncu_FinBressoud_Bailey)