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 $\theta \ge 0$, and establish the same property for the moment sequences associated with all tails of the corresponding Jacobi matrix. Their $\gamma$-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 $\theta >-1$, 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 $\theta \ge 0$, and that every $d$-Hoggatt transformation has the same property for integers $d\ge 1$. The preservation proofs employ the Zhu--Sun criterion and a mixed-minor one-sign-change argument based on a hypergeometric hitting-probability inequality.

arXiv comment22 pages