q-Hypergeometric proof of the key identity for Andrews’ refinement of the Alladi–Schur theorem

Establish a q-hypergeometric proof of identity (6.8), the key identity associated with Andrews’ refinement of the Alladi–Schur theorem for partitions into odd parts occurring no more than twice and Schur-type partitions classified by parity.

Background

Identity (6.8) is obtained by combining Kursungoz’s series representation for Schur partitions, with odd and even parts tracked separately, and Andrews’ refinement of the Alladi–Schur theorem. Its right-hand side is the generating product for partitions into odd parts with multiplicity at most two, while the left-hand side is a multivariate q-series whose coefficients are manifestly nonnegative.

The paper notes that Andrews–Chern–Li later supplied q-hypergeometric and computer-aided proofs of a related identity (6.10). However, the authors specifically emphasize that a q-hypergeometric proof of the nonnegative-coefficient formulation (6.8) was not known at the time of writing. Consequently, proving such a formulation directly remains an explicit problem in the paper.