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Lattice Paths and Branched Continued Fractions: An Infinite Sequence of Generalizations of the Stieltjes–Rogers and Thron–Rogers Polynomials, with Coefficientwise Hankel-Total Positivity
About this Title
Mathias Pétréolle, Alan D. Sokal and Bao-Xuan Zhu
Publication: Memoirs of the American Mathematical Society
Publication Year:
2023; Volume 291, Number 1450
ISBNs: 978-1-4704-6268-0 (print); 978-1-4704-7683-0 (online)
DOI: https://doi.org/10.1090/memo/1450
Published electronically: November 8, 2023
Keywords: Dyck path,
$m$-Dyck path,
Schröder path,
$m$-Schröder path,
Motzkin path,
Łukasiewicz path,
Catalan numbers,
Fuss–Catalan numbers,
Schröder numbers,
continued fraction,
S-fraction,
T-fraction,
J-fraction,
branched continued fraction,
Stieltjes–Rogers polynomials,
Thron–Rogers polynomials,
Jacobi–Rogers polynomials,
production matrix,
totally positive matrix,
total positivity,
Hankel matrix,
Lindström–Gessel–Viennot lemma,
Stieltjes moment problem,
Fuss–Narayana polynomial,
Fuss–Narayana symmetric function,
Eulerian polynomial,
Eulerian symmetric function,
Stirling permutation,
hypergeometric series,
basic hypergeometric series,
contiguous relation
Table of Contents
Chapters
- 1. Introduction
- 2. The $\boldsymbol {m}$-Stieltjes–Rogers and $\boldsymbol {m}$-Thron–Rogers polynomials
- 3. Relation between different values of $\boldsymbol {m}$
- 4. The $\boldsymbol {m}$-Jacobi–Rogers polynomials
- 5. The generalized $\boldsymbol {m}$-Stieltjes–Rogers, $\boldsymbol {m}$-Thron–Rogers and $\boldsymbol {m}$-Jacobi–Rogers polynomials
- 6. Generalized $\boldsymbol {m}$-Jacobi–Rogers polynomials in terms of ordered trees and forests
- 7. Contraction formulae for $\boldsymbol {m}$-branched continued fractions
- 8. Production matrices
- 9. Total positivity
- 10. Weights periodic of period $\boldsymbol {m\!+\!1}$ or $\boldsymbol {m}$
- 11. Weights eventually periodic of period $\boldsymbol {m\!+\!1}$ or $\boldsymbol {m}$
- 12. Weights quasi-affine or factorized of period ${\boldsymbol {m\!+\!1}}$ or $\boldsymbol {m}$
- 13. Ratios of contiguous hypergeometric series I: $\boldsymbol { {{}_{m+1 \!}{F}{_{0}}\!} }$
- 14. Ratios of contiguous hypergeometric series II: $\boldsymbol { {{}_{r }{F}{_{s}}\!} }$
- 15. Ratios of contiguous hypergeometric series III: $\boldsymbol { {_{r}{\phi }{_{s}}} }$
- 16. Some final remarks
Abstract
We define an infinite sequence of generalizations, parametrized by an integer $m \ge 1$, 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 $m$th-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 ${}_r \! \phi _s$.- Martin Aigner, Catalan-like numbers and determinants, J. Combin. Theory Ser. A 87 (1999), no. 1, 33–51. MR 1698277, DOI 10.1006/jcta.1998.2945
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