scholarly journals A Combinatorial Derivation with Schröder Paths of a Determinant Representation of Laurent Biorthogonal Polynomials

10.37236/800 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Shuhei Kamioka
Keyword(s):  
Recurrence Equation ◽  
Combinatorial Proof ◽  
Toeplitz Determinants ◽  
Determinant Representation ◽  
Combinatorial Derivation ◽  
Schröder Paths ◽  
Biorthogonal Polynomials ◽  
Weighted Plane

A combinatorial proof in terms of Schröder paths and other weighted plane paths is given for a determinant representation of Laurent biorthogonal polynomials (LBPs) and that of coefficients of their three-term recurrence equation. In this process, it is clarified that Toeplitz determinants of the moments of LBPs and their minors can be evaluated by enumerating certain kinds of configurations of Schröder paths in a plane.

scholarly journals A Combinatorial Representation with Schröder Paths of Biorthogonality of Laurent Biorthogonal Polynomials

10.37236/955 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Shuhei Kamioka
Keyword(s):  
Linear Functional ◽  
Schröder Paths ◽  
Biorthogonal Polynomials ◽  
Combinatorial Representation ◽  
Weighted Plane

Combinatorial representation in terms of Schröder paths and other weighted plane paths are given of Laurent biorthogonal polynomials (LBPs) and a linear functional with which LBPs have orthogonality and biorthogonality. Particularly, it is clarified that quantities to which LBPs are mapped by the corresponding linear functional can be evaluated by enumerating certain kinds of Schröder paths, which imply orthogonality and biorthogonality of LBPs.

scholarly journals Touchard-Riordan formulas, T-fractions, and Jacobi's triple product identity

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Matthieu Josuat-Vergès ◽  
Jang-Soo Kim
Keyword(s):  
Functional Equation ◽  
Triple Product ◽  
Combinatorial Proof ◽  
Jacobi’S Triple Product Identity ◽  
Product Identity ◽  
Nous Montrons ◽  
Schröder Paths ◽  
International Audience ◽  
Finite Sum ◽  
Jacobi's Triple Product Identity

International audience We give a combinatorial proof of a Touchard-Riordan-like formula discovered by the first author. As a consequence we find a connection between his formula and Jacobi's triple product identity. We then give a combinatorial analog of Jacobi's triple product identity by showing that a finite sum can be interpreted as a generating function of weighted Schröder paths, so that the triple product identity is recovered by taking the limit. This can be stated in terms of some continued fractions called T-fractions, whose important property is the fact that they satisfy some functional equation. We show that this result permits to explain and generalize some Touchard-Riordan-like formulas appearing in enumerative problems. Nous donnons une preuve combinatoire d'une formule à la Touchard-Riordan due au premier auteur. En conséquence, nous faisons appara\^ıtre un lien entre cette formule et l'identité du produit triple de Jacobi. Nous donnons un analogue combinatoire à l'identité du produit triple en montrant qu'une somme finie peut être interprétée comme fonction génératrice de chemins de Schröder pondérés, de sorte que l'identité du produit triple s'obtient en passant à la limite. Ceci peut être énoncé en termes de fractions continues appelées T-fractions, dont la propriété importante est le fait qu'elle satisfont certaines équations fonctionnelles. Nous montrons que ce résultat permet d'expliquer et généraliser certaines formules à la Touchard-Riordan apparaissant dans des problèmes d'énumération.

A NEW COMBINATORIAL STUDY OF THE ROGERS–FINE IDENTITY AND A RELATED PARTIAL THETA SERIES

2009 ◽  
Vol 05 (07) ◽  
pp. 1311-1320 ◽  
Author(s):  
KRISHNASWAMI ALLADI
Keyword(s):  
Theta Series ◽  
Combinatorial Proof ◽  
Combinatorial Derivation

We provide a transparent combinatorial derivation of a variant of the Rogers–Fine identity and a new combinatorial proof of a related partial theta series.

scholarly journals Combinatorial Proofs of Addition Formulas

10.37236/4793 ◽  
2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Xiang-Ke Chang ◽  
Xing-Biao Hu ◽  
Hongchuan Lei ◽  
Yeong-Nan Yeh
Keyword(s):  
Hankel Matrix ◽  
Direct Consequence ◽  
Combinatorial Proof ◽  
Addition Formula ◽  
Polynomial Sequence ◽  
Schröder Paths ◽  
Motzkin Paths ◽  
Addition Formulas

In this paper we give a combinatorial proof of an addition formula for weighted partial Motzkin paths. The addition formula allows us to determine the $LDU$ decomposition of a Hankel matrix of the polynomial sequence defined by weighted partial Motzkin paths. As a direct consequence, we get the determinant of the Hankel matrix of certain combinatorial sequences. In addition, we obtain an addition formula for weighted large Schröder paths.

scholarly journals Multiple phases in a generalized Gross-Witten-Wadia matrix model

2020 ◽  
Vol 2020 (9) ◽  
Author(s):  
Jorge G. Russo ◽  
Miguel Tierz
Keyword(s):  
Partition Function ◽  
Matrix Models ◽  
Characteristic Polynomial ◽  
Matrix Model ◽  
Phase Structure ◽  
Unitary Matrix ◽  
Two Dimensional ◽  
Toeplitz Determinants ◽  
Dimensional Parameter ◽  
Planar Limit

Abstract We study a unitary matrix model of the Gross-Witten-Wadia type, extended with the addition of characteristic polynomial insertions. The model interpolates between solvable unitary matrix models and is the unitary counterpart of a deformed Cauchy ensemble. Exact formulas for the partition function and Wilson loops are given in terms of Toeplitz determinants and minors and large N results are obtained by using Szegö theorem with a Fisher-Hartwig singularity. In the large N (planar) limit with two scaled couplings, the theory exhibits a surprisingly intricate phase structure in the two-dimensional parameter space.

scholarly journals COMBINATORIAL KNOT FLOER HOMOLOGY AND DOUBLE BRANCHED COVERS

2013 ◽  
Vol 22 (06) ◽  
pp. 1350014
Author(s):  
FATEMEH DOUROUDIAN
Keyword(s):  
Floer Homology ◽  
Combinatorial Proof ◽  
Branched Covers ◽  
Knot Floer Homology ◽  
Branched Cover ◽  
Heegaard Diagram

Using a Heegaard diagram for the pullback of a knot K ⊂ S3 in its double branched cover Σ2(K), we give a combinatorial proof for the invariance of the associated knot Floer homology over ℤ.

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