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Askey--Wilson polynomials with ASEP parameters
Abstract
Koornwinder moments generalize the Askey--Wilson moments arising in the asymmetric simple exclusion process. Rains conjectured that, under the specialization t=q, the minimal numerators of Koornwinder moments have nonnegative integer coefficients. While the one-row case of this conjecture was previously proved by Corteel, Mandelshtam, and Williams using rhombic staircase tableaux, its dual counterpart, the one-column case, has remained open. In this paper, we derive a closed, manifestly positive combinatorial formula for the normalized numerators of the coefficients of the rescaled Askey--Wilson polynomials. This proves Rains' conjecture for one-column partitions, thereby providing the exact dual counterpart to the previous result.
Related
- Elementary Askey-Wilson Functions
- The Askey-Wilson Function is the Laplace Transform of a Positive Measure
- Bernstein-type bases on q-quadratic lattices and Askey--Wilson\slash q-Racah connection coefficients
- Irreducibility of Generalised Laguerre Polynomials for Large Integer Parameters
- Limit profile for the ASEP with one open boundary
BibTeX
@misc{cho2026askeywilson,
title = {{Askey--Wilson polynomials with ASEP parameters}},
author = {Younggwang Cho and Donghyun Kim and Jang Soo Kim},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24091},
eprint = {2609.24091},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24091},
}
RIS
TY - UNPB
TI - Askey--Wilson polynomials with ASEP parameters
AU - Younggwang Cho
AU - Donghyun Kim
AU - Jang Soo Kim
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.24091
UR - https://arxiv.org/abs/2609.24091
AB - Koornwinder moments generalize the Askey--Wilson moments arising in the asymmetric simple exclusion process. Rains conjectured that, under the specialization t=q, the minimal numerators of Koornwinder moments have nonnegative integer coefficients. While the one-row case of this conjecture was previously proved by Corteel, Mandelshtam, and Williams using rhombic staircase tableaux, its dual counterpart, the one-column case, has remained open. In this paper, we derive a closed, manifestly positive combinatorial formula for the normalized numerators of the coefficients of the rescaled Askey--Wilson polynomials. This proves Rains' conjecture for one-column partitions, thereby providing the exact dual counterpart to the previous result.
ER -