Schröder matrix as inverse of Delannoy matrix

Yang, Sheng-Liang; Zheng, Siming; He, Tian-Xiao

doi:10.1016/j.laa.2013.09.044

Cited by 33 publications

(11 citation statements)

References 15 publications

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“…Given a Riordan array M = (g(x), f (x)) with matrix representation (t n,k ) we shall denote by its vertical half the matrix V with general (n, k)-th term t 2n−k,n . We have the following result [1,8,9,10].…”

Section: The Vertical and Horizontal Halves Of A Riordan Arraymentioning

confidence: 91%

A Note on Riordan Arrays with Catalan Halves

Barry

2019

Preprint

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For a lower triangular matrix (t n,k ) we call the matrices with respective entries (t 2n−k,n ) and (t 2n,n+k ) the vertical and the horizontal halves. In this note, we discuss Riordan arrays whose halves are closely related to the Catalan matrices. Preliminaries on Riordan arraysWe recall some facts about Riordan arrays in this introductory section. Readers familiar with Riordan arrays may wish to move on to the next section.A Riordan array [2, 5] is defined by a pair of power series

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Section: The Vertical and Horizontal Halves Of A Riordan Arraymentioning

confidence: 91%

A Note on Riordan Arrays with Catalan Halves

Barry

2019

Preprint

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“…Given a Riordan array M = (g(x), f (x)) = (g(x), xh(x)) with matrix representation (T n,k ) we shall denote by its vertical half the matrix M V with general (n, k)-th term T 2n−k,n . We have the following result [7,8].…”

Section: The Vertical and Horizontal Halves Of A Riordan Arraymentioning

confidence: 92%

On the halves of a Riordan array and their antecedents

Barry

2019

Linear Algebra and its Applications

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Every Riordan array has what we call a horizontal half and a vertical half. These halves of a Riordan array have been studied separately before. Here, we place them in a common context, showing that one may be obtained from the other. Using them, we provide a canonical factorization of elements of the associated or Lagrange subgroup of the Riordan group. The vertical half matrix is shown to be an element of the hittingtime group. We also ask and answer the question: given a Riordan array, when is it the half (either horizontal of vertical) of a Riordan array?

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“…. Therefore, by Theorem 2.5, the (m, r)-central coefficient triangle of the Pascal matrix G = (1/(1 − t), t/(1 − t)) can be written as P r o o f. By (22), the generic term of (B m (t) r , tB…”

Section: P R O O F For the Riordan Arraymentioning

confidence: 97%