Continuous Narayana Interpolations: Hankel Total Positivity and Log-Convexity Preservation
Abstract
We study a continuous interpolation between the type B and shifted type A Narayana triangles. Using a direct Gegenbauer-moment proof and a continuant comparison criterion, we show that the row polynomials form a coefficientwise Hankel-totally positive sequence for every real θ≥0, and establish the same property for the moment sequences associated with all tails of the corresponding Jacobi matrix. Their γ-expansion, real-rootedness and Stieltjes moment representation are explicit, as are the exact orders of vanishing and extreme coefficients of arbitrary Hankel minors. This confirms and strengthens Rodelet--Causse--Tevlin's log-convexity conjecture for super Narayana polynomials. We further extend Wang--Yang's total positivity of the type~A and B Narayana triangles and strict total positivity of the associated squares to every real θ>−1, using Pólya-frequency sequences and a gamma-product moment representation. We also prove that the continuous Narayana transformations preserve strong t-log-convexity for every θ≥0, and that every d-Hoggatt transformation has the same property for integers d≥1. The preservation proofs employ the Zhu--Sun criterion and a mixed-minor one-sign-change argument based on a hypergeometric hitting-probability inequality.
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BibTeX
@misc{zeng2026continuous,
title = {{Continuous Narayana Interpolations: Hankel Total Positivity and Log-Convexity Preservation}},
author = {Jiang Zeng},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.19548},
eprint = {2610.19548},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.19548},
}
RIS
TY - UNPB
TI - Continuous Narayana Interpolations: Hankel Total Positivity and Log-Convexity Preservation
AU - Jiang Zeng
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.19548
UR - https://arxiv.org/abs/2610.19548
AB - We study a continuous interpolation between the type B and shifted type A Narayana triangles. Using a direct Gegenbauer-moment proof and a continuant comparison criterion, we show that the row polynomials form a coefficientwise Hankel-totally positive sequence for every real θ≥0, and establish the same property for the moment sequences associated with all tails of the corresponding Jacobi matrix. Their γ-expansion, real-rootedness and Stieltjes moment representation are explicit, as are the exact orders of vanishing and extreme coefficients of arbitrary Hankel minors. This confirms and strengthens Rodelet--Causse--Tevlin's log-convexity conjecture for super Narayana polynomials. We further extend Wang--Yang's total positivity of the type~A and B Narayana triangles and strict total positivity of the associated squares to every real θ>−1, using Pólya-frequency sequences and a gamma-product moment representation. We also prove that the continuous Narayana transformations preserve strong t-log-convexity for every θ≥0, and that every d-Hoggatt transformation has the same property for integers d≥1. The preservation proofs employ the Zhu--Sun criterion and a mixed-minor one-sign-change argument based on a hypergeometric hitting-probability inequality.
ER -