Some Matrix Identities on Colored Motzkin Paths

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

Research output: Journal ArticleArticlepeer-review


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 languageAmerican English
JournalDiscrete Mathematics
StatePublished - Dec 2017


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


  • Applied Mathematics
  • Mathematics

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