Two q-Deformations of Narayana Polynomials: Rogers Moments, Hankel Positivity, and q-Narayana Kernels
Year: 2026
Type: preprint
Abstract: We study two $q$-deformations of the continuous interpolation between type $B$ and shifted type $A$ Narayana polynomials, both constructed from the continuous $q$-ultraspherical (Rogers) functional. The first consists of affine moments of ordinary powers. For fixed $0-1$, we prove total positivity of the kernel triangle by a Jackson $q$-Bessel--Edrei factorization and strict total positivity of its associated square by a discrete moment representation. This gives a simultaneous $(θ,q)$-extension of Wang--Yang's theorems for type $A$ and type $B$ Narayana triangles and squares. We further establish coefficientwise total nonnegativity in $q$ through order two for the endpoint triangles. An exact triangular change of basis links the two deformations and explains their common classical limit.
Source: arXiv (Cornell University)
Author: Zeng, Jiang
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223203774