Lattice paths and branched fractions : an infinite sequence of generalizations of the Stieltjes-Rogers and Thron-Rogers polynomials, with coefficientwise Hankel-total positivity / Mathias Pétréolle, Alan D. Sokal, Bao-Xuan Zhu.

Format:

Book

Author/Creator:

Pétréolle, Mathias, author.

Sokal, Alan D., 1955- author.

Zhu, Bao-Xuan, author.

Series:

Memoirs of the American Mathematical Society ; v. 1450.

Memoirs of the American Mathematical Society, 0065-9266 ; no. 1450

Language:

English

Subjects (All):

Lattice paths.

Polynomials.

Fractions.

Physical Description:

v, 154 pages : illustrations ; 26 cm.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2023]

Summary:

"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-Schr¨oder 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 rFs for arbitrary r and s, which generalize Gauss' continued fraction for ratios of contiguous 2F1; and for s = 0 we prove the coefficientwise Hankel-total positivity. Finally, we extend the branched continued fractions to ratios of contiguous basichypergeometric series rφs." -- Provided by publisher

Contents:

Introduction

The m-Stieltjes-Rogers and m-Thron-Rogers polynomials

Relation between different values of m

The m-Jacobi-Rogers polynomials

The generalized m-Stieltjes-Rogers polynomials in terms of ordered trees and forests

Contracttion formulae for m-branched continued fractions

Total positivity

Weights periodic of period m+1 or m

Weights eventually periodic of period m+1 or m

Weights quasi-affine or factorize of period m +1 or m

Ratios of contiguous hypergeometric series I: m+1Fo

Ratios of contiguous hypergeometric series II: rF8

Ratios of contiguous hypergeometric series III: rOs

Some final remarks.

Notes:

"November 2023, volume 291, number 1450 (fifth of 5 numbers)."

Includes bibliographical references (pages 147-154).

ISBN:

1470462680

9781470462680

OCLC:

1413943532