{
  "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 $q^2$ 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.28691",
  "author_total": 1,
  "authors": [
    {
      "name": "Jiang Zeng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/59bjrsm9x6b46.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.28691",
  "fulltext_url": null,
  "id": "59bjrsm9x6b46",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.28691",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7223203774",
  "pdf_url": "https://arxiv.org/pdf/2610.28691",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/59bjrsm9x6b46.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:22Z",
      "native_id": "2610.28691",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.28691"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Two q-Deformations of Narayana Polynomials: Rogers Moments, Hankel Positivity, and q-Narayana Kernels",
  "title_text": "Two q-Deformations of Narayana Polynomials: Rogers Moments, Hankel Positivity, and q-Narayana Kernels",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:22Z",
  "url": "https://research.draconitic.com/paper/59bjrsm9x6b46",
  "venue": "arXiv",
  "versions": []
}