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Finding an Isomorphism between the Riordan Group and a Subgroup of the Double Riordan Group

Shakuan Frankson

Abstract

The Riordan group is a set of infinite lower-triangular matrices defined by two generating functions, $g$ and $f$. The elements of the group are called Riordan arrays, denoted by $(g,f)$, and the $k$th column of a Riordan array is given by the function $gf^k$. The Double Riordan group is defined similarly using three generating functions $g$, $f_1$, and $f_2$, where $g$ is an even function and $f_1$ and $f_2$ are odd functions. This group generalizes the Checkerboard subgroup of the Riordan group, where $g$ is even and $f$ is odd. An open question posed by Davenport, Shapiro, and Woodson was if there exists an isomorphism between the Riordan group and a subgroup of the Double Riordan group. This question is answered in this article.

Finding an Isomorphism between the Riordan Group and a Subgroup of the Double Riordan Group

Abstract

The Riordan group is a set of infinite lower-triangular matrices defined by two generating functions, and . The elements of the group are called Riordan arrays, denoted by , and the th column of a Riordan array is given by the function . The Double Riordan group is defined similarly using three generating functions , , and , where is an even function and and are odd functions. This group generalizes the Checkerboard subgroup of the Riordan group, where is even and is odd. An open question posed by Davenport, Shapiro, and Woodson was if there exists an isomorphism between the Riordan group and a subgroup of the Double Riordan group. This question is answered in this article.

Paper Structure

This paper contains 4 sections, 10 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. The Isomorphism Problem
  3. The Monomorphism Theorems
  4. Conclusion

Key Result

Theorem 1

(The Fundamental Theorem of Riordan Arrays): Let $A(z)=\sum_{k=0}^{\infty }{a_{k}z^{k}}$ and $B(z)=\sum_{k=0}^{\infty }{b_{k}z^{k}}$ and let $A$ and $B$ be the column vectors $A=\left( a_{0},a_{1},a_{2},\cdots \right) ^{T}$ and $B=\left( b_{0},b_{1},b_{2},\cdots \right) ^{T}$. Then $(g,f)*A=B$, if a

Theorems & Definitions (23)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1
  • proof
  • Theorem 2
  • Definition 4
  • Theorem 3
  • Theorem 4
  • Lemma 1
  • ...and 13 more