Some Matrix Identities on Colored Motzkin Paths

Tian-Xiao He; Sheng-liang Yang; Yan-Ni Dong

doi:10.1016/j.disc.2017.07.006

Some Matrix Identities on Colored Motzkin Paths

Tian-Xiao He, Sheng-liang Yang, Yan-Ni Dong

Illinois Wesleyan University

Research output: Journal Article › Article › peer-review

Abstract

Merlini and Sprugnoli (2017) give both an algebraic and a combinatorial proof for an identity proposed by Louis Shapiro by using Riordan arrays and a particular model of lattice paths. In this paper, we revisit the identity and emphasize the use of colored partial Motzkin paths as appropriate tool. By using colored Motzkin paths with weight defined according to the height of its last point, we can generalize the identity in several ways. These identities allow us to move from Fibonacci polynomials, Lucas polynomials, and Chebyshev polynomials, to the polynomials of the form ( z + b )n.

Original language	American English
Journal	Discrete Mathematics
Volume	340
DOIs	https://doi.org/10.1016/j.disc.2017.07.006
State	Published - Dec 2017

Keywords

Catalan matrix
Chebyshev polynomial
Fibonacci polynomial
Lucas polynomial
Motzkin path
Riordan array

Disciplines

Applied Mathematics
Mathematics

Access to Document

10.1016/j.disc.2017.07.006License: Unspecified

Cite this

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title = "Some Matrix Identities on Colored Motzkin Paths",

abstract = " Merlini and Sprugnoli (2017) give both an algebraic and a combinatorial proof for an identity proposed by Louis Shapiro by using Riordan arrays and a particular model of lattice paths. In this paper, we revisit the identity and emphasize the use of colored partial Motzkin paths as appropriate tool. By using colored Motzkin paths with weight defined according to the height of its last point, we can generalize the identity in several ways. These identities allow us to move from Fibonacci polynomials, Lucas polynomials, and Chebyshev polynomials, to the polynomials of the form ( z + b )n.",

keywords = "Catalan matrix, Chebyshev polynomial, Fibonacci polynomial, Lucas polynomial, Motzkin path, Riordan array",

author = "Tian-Xiao He and Sheng-liang Yang and Yan-Ni Dong",

year = "2017",

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doi = "10.1016/j.disc.2017.07.006",

language = "American English",

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journal = "Discrete Mathematics",

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AB - Merlini and Sprugnoli (2017) give both an algebraic and a combinatorial proof for an identity proposed by Louis Shapiro by using Riordan arrays and a particular model of lattice paths. In this paper, we revisit the identity and emphasize the use of colored partial Motzkin paths as appropriate tool. By using colored Motzkin paths with weight defined according to the height of its last point, we can generalize the identity in several ways. These identities allow us to move from Fibonacci polynomials, Lucas polynomials, and Chebyshev polynomials, to the polynomials of the form ( z + b )n.

KW - Catalan matrix

KW - Chebyshev polynomial

KW - Fibonacci polynomial

KW - Lucas polynomial

KW - Motzkin path

KW - Riordan array

U2 - 10.1016/j.disc.2017.07.006

DO - 10.1016/j.disc.2017.07.006

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JO - Discrete Mathematics

JF - Discrete Mathematics

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