Two q-Deformations of Narayana Polynomials: Rogers Moments, Hankel Positivity, and q-Narayana Kernels
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<q<1 and −1<β<1, we prove coefficientwise Hankel total positivity in the polynomial variable by a continuant comparison criterion. We also determine leading continuants, Hankel determinants, and extreme coefficients of arbitrary Hankel minors. The second deformation uses Askey--Wilson basis products and yields a basic-hypergeometric Narayana kernel. We give two independent moment evaluations and prove that its row polynomials have simple negative zeros and are γ-positive after palindromic normalization. An Askey--Wilson integral with base q2 gives a continuous-parameter extension and a moment proof of the Krattenthaler--Wachs super q-Narayana expansion, including its integrality. An ordinary-power expansion also proves Pan's Touchard identity. For fixed 0<q<1 and θ>−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.
Related
- Riesz kernels of hyperbolic polynomials: positivity, admissible exponents and Jordan rigidity
- Some sequences and number triangles which are related to Narayana polynomials and to q-Narayana polynomials for q=-1
- Zeros and interlacing for multiset Eulerian-Narayana polynomials
- Continuous Narayana Interpolations: Hankel Total Positivity and Log-Convexity Preservation
- Pfaffian Positivity for Convex Kernels and Optimal Matching Bounds
BibTeX
@misc{zeng2026qdeformations,
title = {{Two q-Deformations of Narayana Polynomials: Rogers Moments, Hankel Positivity, and q-Narayana Kernels}},
author = {Jiang Zeng},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28691},
eprint = {2610.28691},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28691},
}
RIS
TY - UNPB
TI - Two q-Deformations of Narayana Polynomials: Rogers Moments, Hankel Positivity, and q-Narayana Kernels
AU - Jiang Zeng
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28691
UR - https://arxiv.org/abs/2610.28691
AB - 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<q<1 and −1<β<1, we prove coefficientwise Hankel total positivity in the polynomial variable by a continuant comparison criterion. We also determine leading continuants, Hankel determinants, and extreme coefficients of arbitrary Hankel minors. The second deformation uses Askey--Wilson basis products and yields a basic-hypergeometric Narayana kernel. We give two independent moment evaluations and prove that its row polynomials have simple negative zeros and are γ-positive after palindromic normalization. An Askey--Wilson integral with base q^2 gives a continuous-parameter extension and a moment proof of the Krattenthaler--Wachs super q-Narayana expansion, including its integrality. An ordinary-power expansion also proves Pan's Touchard identity. For fixed 0<q<1 and θ>−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.
ER -