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The Vertical Recursive Relation of Riordan Arrays and Its Matrix Representation
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Tian-Xiao He

Department of Mathematics

Illinois Wesleyan University

Bloomington, IL 61702-2900

USA

**Abstract:**

It is known that the entries of a Riordan array satisfy horizontal
recursive relations represented by the *A*- and *Z*-sequences.
In this
paper, we study a vertical recursive relation approach to Riordan arrays.
This vertical recursive approach gives a way to represent the entries of
a Riordan array (*g*, *f*)
in terms of a recursive linear combination of the
coefficients of *g*. We also give a matrix representation of the vertical
recursive relation. The set of all those matrices forms a group, called
the quasi-Riordan group. We present extensions of the horizontal
recursive relation and the vertical recursive relation in terms of *c*-
and *C*-Riordan arrays, with illustrations by using the rook triangle
and the Laguerre triangle. These extensions represent a way to study
nonlinear recursive relations of the entries of some triangular matrices
from linear recursive relations of the entries of Riordan arrays. In
addition, the matrix representation of the vertical recursive relation of
Riordan arrays provides transforms between lower order and higher order
finite Riordan arrays, where the *m*th order Riordan array is defined
by (*g*, *f*)_{m} = (*d*_{n,k})_{m ≥ n,k ≥ 0}. Furthermore, the vertical
relation approach to Riordan arrays provides a unified approach to
construct identities.

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(Concerned with sequence
A001764.)

Received September 30 2022; revised versions received October 1 2022; October 3 2022; November 14 2022; November 15 2022; November 16 2022; November 17 2022.
Published in *Journal of Integer Sequences*,
November 18 2022.

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