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, Mathias; Sokal, Alan D.; Zhu, Bao-Xuan

doi:10.48550/arxiv.1807.03271

Cited by 15 publications

(72 citation statements)

References 94 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For example, the Bell polynomials, the classical Eulerian polynomials, the Narayana polynomials of type A and B, Ramanujan polynomials, Dowling polynomials, Jacobi-Stirling polynomials, and so on, are q-log-convex (see Chen et al [20,21], Liu and Wang [58], Zhu [103,104,105,106], Zhu and Sun [113] for instance), 3-q-log-convex (see [107]) and q-Stieltjes moment (see [90,101,107,109]). We refer the reader to [73,74,90,91,110,111,112] for coefficientwise Hankel-total positivity in more indeterminates.…”

Section: Definitions and Notation From Total Positivitymentioning

confidence: 99%

“…Recently, in [74], based on lattice paths and branched continued fractions, Pétréolle, Sokal and Zhu developed the theory for the coefficientwise Hankel-total positivity of the m-Stieltjes-Rogers polynomial sequence in all the indeterminates. Note that an m-Stieltjes-Rogers polynomial sequence coincides with the zeroth column of certain m-Jacobi-Rogers triangle.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Zhu¹

2022

Preprint

Self Cite

View full text Add to dashboard Cite

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

show abstract

Section: Definitions and Notation From Total Positivitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Zhu¹

2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…The coefficients above were obtained from [29,Th. 14.5] as the coefficients of a branched continued fraction representation for 3 F 2 (a, b, 1; c, d;t), the ordinary generating function of the moment sequence given by (1.5).…”

Section: Recurrence Relationmentioning

confidence: 99%

“…For instance, the analysis of singular values of products of Ginibre matrices in [19,18] uses multiple orthogonal polynomials associated with weight functions expressed in terms of Meijer G-functions, a class of weights to which the weight (1.1) belongs. Besides, these polynomials are linked with the branched continued fractions introduced in [29] as the generating functions of m-Dyck paths, for the purpose of solving total positivity problems involving combinatorially interesting sequences of polynomials. This connection, which leads to new results on both fields involved, will be further explored in forthcoming work.…”

Section: Introductionmentioning

confidence: 99%

“…So, in §3.5 we make the connection to the so-called Jacobi-type 2-orthogonal polynomials investigated in [21], where we pay particular attention to the case where all the coefficients are constant. The recurrence relation coefficients of the multiple orthogonal polynomials under analysis can be written as combinations of the coefficients of a branched continued fraction representation for a generalised hypergeometric function derived in [29]. Hence, these recurrence coefficients are real, positive and bounded, whose asymptotic behaviour coincides with the one of the recurrence relation coefficients of Jacobi-Piñeiro (type II) multiple orthogonal polynomials on the step line studied in [6].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Multiple orthogonal polynomials with respect to Gauss' hypergeometric function

Lima¹,

Loureiro²

2020

Preprint

View full text Add to dashboard Cite

A new set of multiple orthogonal polynomials of both type I and type II with respect to two weight functions involving Gauss' hypergeometric function on the interval (0, 1) is studied. This type of polynomials have direct applications in the investigation of singular values of products of Ginibre matrices, in the analysis of rational solutions to Painlevé equations and are connected with branched continued fractions and total positivity problems in combinatorics. The pair of orthogonality measures is shown to be a Nikishin system and to satisfy a matrix Pearson-type differential equation. The focus is on the polynomials whose indexes lie on the step line, for which it is shown that a differentiation on the variable gives a shift on the parameters, therefore satisfying the Hahn's property. We obtain a Rodrigues-type formula for the type I, while a more detailed characterisation is given for the type II polynomials (aka 2-orthogonal polynomials) which includes: an explicit expression as a terminating hypergeometric series, a third-order differential equation and a third-order recurrence relation. The asymptotic behaviour of their recurrence coefficients mimics those of Jacobi-Piñeiro polynomials, based on which, their zero asymptotic distribution and a Mehler-Heine asymptotic formula near the origin are given. Particular choices on the parameters degenerate in some known systems such as special cases of the Jacobi-Piñeiro polynomials, Jacobi-type 2-orthogonal polynomials and components of the cubic decomposition of threefold symmetric Hahn-classical polynomials. Equally considered are confluence relations to other known polynomial sets, such as multiple orthogonal polynomials with respect to Tricomi functions.

show abstract

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

Sokal

2022

Monatsh Math

View full text Add to dashboard Cite

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.

show abstract

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

Cited by 15 publications

References 94 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

Multiple orthogonal polynomials with respect to Gauss' hypergeometric function

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

Contact Info

Product

Resources

About