The Minimal Denominator of Dual Koornwinder Coefficients

Abstract

Let N^Z_lambda denote the dual coefficients in the Schur expansion of the rescaled multivariate Askey-Wilson polynomials at the Koornwinder specialization t = q. Cho, Kim, Kim, Lee, Liu and Song conjectured an explicit cell-product formula for the minimal denominator of N^Z_lambda. We prove this formula for every partition lambda in Par_n over the generic parameter field. Starting from the factorial Askey-Wilson coefficient formula of Corteel, Jonnadula, Keating and Kim, we factor the relevant minors into row and column terms and a determinant of monic polynomial evaluations. Vandermonde cancellation together with the beta-set complement identity gives the candidate denominator, including repeated factors. To prove minimality rather than only an upper bound, we introduce a one-parameter deformation. A unique maximal-degree contribution in the resulting Cauchy-Binet expansion has a nonzero coefficient independent of the denominator parameter, excluding further generic cancellation. The affine rescaling and ASEP parametrization transfer the pole multiplicities to the predicted factors. This proves Conjecture 4.17 and is complementary to the recent one-column positivity result of Cho, Kim and Kim (arXiv:2609.24091v1). Manuscript version v0.6. Zenodo public version v2. Published as a preprint on 22 September 2026. Not peer reviewed.