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arXiv · 2610.28691

Two q-Deformations of Narayana Polynomials: Rogers Moments, Hankel Positivity, and q-Narayana Kernels

Jiang Zeng
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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.

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Jiang Zeng. 2026-10-20. Two q-Deformations of Narayana Polynomials: Rogers Moments, Hankel Positivity, and q-Narayana Kernels. https://arxiv.org/abs/2610.28691

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