Row polynomial matrices of Riordan arrays

Mu, Lili; Mao, Jianxi; Wang, Yi

doi:10.1016/j.laa.2017.02.006

Cited by 9 publications

(3 citation statements)

References 26 publications

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“…where J n,k = 0 unless 0 ≤ k ≤ n and J 0,0 = 1. The x-log-convexity, x-log-convexity of higher order and coefficientwise Hankel-total positivity of (J n,0 ) n≥0 have been studied, see [18,69,70,91,101,103,105,107,110] for instance. But the following question is open.…”

Section: Row-generating Polynomials For Catalan-stieltjes Matricesmentioning

confidence: 99%

Coefficientwise Hankel-total positivity of row-generating polynomials for the $m$-Jacobi-Rogers triangle

Zhu¹

2022

Preprint

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Total positivity of matrices is deeply studied and plays an important role in various branches of mathematics. Recently, based on lattice paths and branched continued fractions, Pétréolle, Sokal and Zhu (Mem. Amer. Math. Soc., to appear) developed the theory for the coefficientwise Hankel-total positivity of the m-Stieltjes-Rogers polynomial sequence in all the indeterminates. The aim of this paper is to study the criteria for the row-generating polynomial sequence of the m-Jacobi-Rogers triangle being coefficientwise Hankel-totally positive and their applications.Using the theory of production matrices, we gain a criterion for the coefficientwise Hankel-total positivity of the row-generating polynomial sequence of the m-Jacobi-Rogers triangle. This immediately implies that the corresponding m-Jacobi-Rogers triangular convolution preserves the Stieltjes moment property of sequences and its zeroth column sequence is coefficientwise Hankel-totally positive and logconvex of higher order in all the indeterminates. In consequence, for m = 1, we immediately obtain some results on Hankel-total positivity for the Catalan-Stieltjes matrices. In particular, we in a unified manner apply our results to some combinatorial triangles or polynomials including the generalized Jacobi Stirling triangle, a generalized elliptic polynomial, a refined Stirling cycle polynomial and a refined Eulerian polynomial. For the general m, combining our criterion and a function satisfying an autonomous differential equation, we present different criteria for coefficientwise Hankel-total positivity of the row-generating polynomial sequence for exponential Rirodan arrays. In addition, we also derive some results for the coefficientwise Hankel-total positivity in terms of compositional functions and m-branched Stieltjes continued fractions. Finally, we apply our criteria to: (1) rook polynomials and signless Laguerre polynomials (confirming a conjecture of Sokal on coefficientwise Hankel-total positivity of rook polynomials), (2) labeled trees and forests (proving some conjectures of Sokal on total positivity and Hankel-total positivity), (3) rthorder Eulerian polynomials (giving a new proof for the coefficientwise Hankel-total

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Section: Row-generating Polynomials For Catalan-stieltjes Matricesmentioning

confidence: 99%

Coefficientwise Hankel-total positivity of row-generating polynomials for the $m$-Jacobi-Rogers triangle

Zhu¹

2022

Preprint

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“…where ξ is an indeterminate. More generally, it will prove useful to define [29,108,165] the row-generating matrix…”

Section: Relation Between Different Values Of Mmentioning

confidence: 99%

Lattice paths and branched continued fractions: An infinite sequence of generalizations of the Stieltjes--Rogers and Thron--Rogers polynomials, with coefficientwise Hankel-total positivity

Pétréolle,

Sokal,

Zhu

2018

Preprint

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We define an infinite sequence of generalizations, parametrized by an integer m ≥ 1, of the Stieltjes-Rogers and Thron-Rogers polynomials; they arise as the power-series expansions of some branched continued fractions, and as the generating polynomials for m-Dyck and m-Schröder paths with height-dependent weights. We prove that all of these sequences of polynomials are coefficientwise Hankel-totally positive, jointly in all the (infinitely many) indeterminates. We then apply this theory to prove the coefficientwise Hankel-total positivity for combinatorially interesting sequences of polynomials. Enumeration of unlabeled ordered trees and forests gives rise to multivariate Fuss-Narayana polynomials and Fuss-Narayana symmetric functions. Enumeration of increasing (labeled) ordered trees and forests gives rise to multivariate Eulerian polynomials and Eulerian symmetric functions, which include the univariate mth-order Eulerian polynomials as specializations. We also find branched continued fractions for ratios of contiguous hypergeometric series r F s for arbitrary r and s, which generalize Gauss' continued fraction for ratios of contiguous 2 F 1 ; and for s = 0 we prove the coefficientwise Hankel-total positivity. Finally, we extend the branched continued fractions to ratios of contiguous basic hypergeometric series r φ s .

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“…Let us remark that the ordinary row-generating matrix of a lower-triangular matrix-that is, (1.5) without the factors k -has been introduced recently by several authors[16,87,131]. I do not know whether the binomial row-generating matrix has been used previously, but I suspect that it has been.…”

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confidence: 99%

Total positivity of some polynomial matrices that enumerate labeled trees and forests I: forests of rooted labeled trees

Sokal

2022

Monatsh Math

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We consider the lower-triangular matrix of generating polynomials that enumerate k-component forests of rooted trees on the vertex set [n] according to the number of improper edges (generalizations of the Ramanujan polynomials). We show that this matrix is coefficientwise totally positive and that the sequence of its row-generating polynomials is coefficientwise Hankel-totally positive. More generally, we define the generic rooted-forest polynomials by introducing also a weight $$m! \, \phi _m$$ m ! ϕ m for each vertex with m proper children. We show that if the weight sequence $$\varvec{\phi }$$ ϕ is Toeplitz-totally positive, then the two foregoing total-positivity results continue to hold. Our proofs use production matrices and exponential Riordan arrays.

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Row polynomial matrices of Riordan arrays

Cited by 9 publications

References 26 publications

Coefficientwise Hankel-total positivity of row-generating polynomials for the $m$-Jacobi-Rogers triangle

Coefficientwise Hankel-total positivity of row-generating polynomials for the $m$-Jacobi-Rogers triangle

Lattice paths and branched continued fractions: An infinite sequence of generalizations of the Stieltjes--Rogers and Thron--Rogers polynomials, with coefficientwise Hankel-total positivity

Total positivity of some polynomial matrices that enumerate labeled trees and forests I: forests of rooted labeled trees

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