Results in Mathematics

, 74:169

# Sequence Characterization of 3-Dimensional Riordan Arrays and Some Application

Open Access
Article

## Abstract

We propose the characterization of 3-dimensional Riordan arrays with use of three sequences that is analogous to the representation of 2-dimensional Riordan arrays with use of A and Z-sequence. We also suggest an application of this representation for finding totally positive matrices.

## Keywords

Riordan group multi-dimensional Riordan array 3-dimensional Riordan array Riordan array sequence characterization A-sequence Z-sequence

## Mathematics Subject Classification

Primary 05A15 15B99 15B05 05A05

## 1 Introduction

Let’s recall that the Riordan group, introduced in , is a group of $$\mathbb {N}_0\times \mathbb {N}_0$$ matrices that are identified with pairs of formal power series. Namely, denoting by $${\mathcal {F}}_0$$—the ring of formal power series with nonzero free term, and by $${\mathcal {F}}_1$$—the ring of formal power series with zero free term but nonzero the next term, the Riordan group $${\mathcal {R}}$$ consists of pairs $${\mathcal {R}}(g,f)$$ with $$g\in {\mathcal {F}}_0$$, $$f\in {\mathcal {F}}_1$$. The multiplication of these pairs is given by
\begin{aligned} {\mathcal {R}}(g_1(z),f_1(z))*{\mathcal {R}}(g_2(z),f_2(z))={\mathcal {R}}(g_1(z)\cdot g_2(f_1(z)),f_2(f_1(z))), \end{aligned}
and it coincides with multiplication of corresponding matrices.
Recently, one can observe an interest in multi-dimensional matrix algebra [2, 3, 4]. Here we would like to focus on three-dimensional matrices. The (2, 1)-product $$C=[c_{ijk}]$$ of $$A=[a_{ijk}]$$, $$B=[b_{ijk}]$$, is defined by the formula:
\begin{aligned} c_{ijk}=\sum _{x\ge 0}a_{ixk}b_{xjk}. \end{aligned}
(1.1)
In this note we are interested in $${\mathcal {R}}^{\langle 3\rangle }$$—the group of 3-dimensional Riordan arrays. It was proved by Cheon and Jin  that $${\mathcal {R}}^{\langle 3\rangle }$$ is an extension of $${\mathcal {R}}$$ by $${\mathcal {F}}_0$$. In this group the matrices $$R=[r_{nkm}]_{n,k,m\in \mathbb {N}_0}$$ are associated with the triples of series (gfh) with $$g,h\in {\mathcal {F}}_0$$, $$f\in {\mathcal {F}}_1$$. The multiplication of such triples is defined as follows:
\begin{aligned} \begin{array}{c} {\mathcal {R}}(g_1(z),f_1(z),h_1(z))*{\mathcal {R}}(g_2(z),f_2(z),h_2(z)) \\ =\\ {\mathcal {R}}(g_1(z)\cdot g_2(f_1(z)),f_2(f_1(z)),h_1(z)\cdot h_2(f_1(z))),\\ \end{array} \end{aligned}
(1.2)
Each entry of $$R=[r_{nkm}]$$ can be found from the relation:
\begin{aligned} r_{nkm}=[z^n]gf^kh^m, \end{aligned}
(1.3)
where $$[z^n]f$$ denotes the n-th coefficient in the series expansion of f.

Thanks to definition (1.3), multiplication (1.2) corresponds with matrix multiplication given by (1.1).

It is known (see [6, 7]) that Riordan arrays can be uniquely determined by two sequences, called A-sequence and Z-sequence. More precisely, starting with $$r_{00}=g_0$$, all the other entries can be found using the relations:
\begin{aligned} r_{n+1,k+1}= & {} a_0r_{nk}+a_1r_{n,k+1}+a_2r_{n,k+2}+\cdots , \nonumber \\ r_{n+1,0}= & {} z_0r_{n0}+z_1r_{n1}+z_2r_{n2}+\cdots \; . \end{aligned}
(1.4)
In this paper we wish to give an analogous presentation for 3-dimensional Riordan arrays. Namely, we propose A, Z, and H-sequence, that completely characterize Riordan array:
\begin{aligned} \begin{aligned} r_{n00}&=z_0r_{n-1,0,0}+z_1r_{n-1,1,0}+z_2r_{n-1,2,0}+\cdots \\ r_{nk0}&=a_0r_{n-1,k+1,0}+a_1r_{n-1,k+2,0}+a_2r_{n-1,k+3,0}+\cdots \\ r_{nkm}&=h_0r_{n,k,m-1}+h_1r_{n-1,k,m-1}+h_2r_{n-2,k,m-1}+\cdots \; . \end{aligned} \end{aligned}
(1.5)
We will show that the following theorem is true.

### Theorem 1.1

Any 3-dimensional Riordan array is completely characterized by its A, Z and H sequence given as in (1.5). Moreover
\begin{aligned} f(z)=z\cdot A(f(z)), \quad g(z)=\frac{g_0}{1-z\cdot Z(f(z))}, \quad H(z)=h(z). \end{aligned}
(1.6)

After discussing the above presentation, we will propose how it can be used to obtain some totally positive Riordan arrays.

## 2 The Disscussion

### 2.1 Sequence Characterization

To prove the main result, it suffices to notice that the below lemma holds.

### Lemma 2.1

The groups
1. 1.

$${\mathcal {R}}_g^{\langle 3\rangle }:=\left\{ (g,f,1):\; g\in {\mathcal {F}}_0,\; f\in {\mathcal {F}}_1\right\}$$,

2. 2.

$${\mathcal {R}}_h^{\langle 3\rangle }:=\left\{ (1,f,h):\; h\in {\mathcal {F}}_0,\; f\in {\mathcal {F}}_1\right\}$$

are isomorphic with $${\mathcal {R}}$$. Moreover
1. 1.

A and Z-sequence of $${\mathcal {R}}(g,f,1)$$ coincide with A and Z-sequence of $${\mathcal {R}}(g,f)$$,

2. 2.

H and Z-squence of $${\mathcal {R}}(1,f,h)$$ coincide with A and Z-sequence of $${\mathcal {R}}(h,f)$$.

### Proof

Clearly, the maps $$\phi _g:\;{\mathcal {R}}_g^{\langle 3\rangle }\rightarrow {\mathcal {R}}$$, $$\phi _h:\;{\mathcal {R}}_h^{\langle 3\rangle }\rightarrow {\mathcal {R}}$$ given by
\begin{aligned} \phi _g({\mathcal {R}}(g,h,1))=(g,h), \qquad \phi _h({\mathcal {R}}(1,f,h))=(h,f) \end{aligned}
establish the desired isomorphism, and the correspondence of sequences. $$\square$$

### Proof of Theorem 1.1

Comparing (1.4) and (1.5) one can notice that A and Z-sequence of $${\mathcal {R}}(g,f,h)$$ coincide with A and Z-sequence of $${\mathcal {R}}(g,f)$$. Thus, we only need to check the last equality. Using (1.3) we get
\begin{aligned} r_{nkm}= & {} [z^n]gf^kh^m =\sum _{i=0}^n([z^i]gf^k\cdot [z^{n-i}]h) =\sum _{i=0}r_{i,k,m-1}\cdot h_{n-i} \\= & {} \sum _{j=0}^nh_jr_{n-j,k,m-1}. \end{aligned}
$$\square$$

For 2-dimensional Riordan arrays the following result was obtained by He and Sprugnoli.

### Theorem 2.2

[8, Thm.3.3,3.4]. Let $${\mathcal {R}}(g_1,f_1)$$, $${\mathcal {R}}(g_2,f_2)$$ be 2-dimensional Riordan arrays with A, Z-sequences: $$A_1$$, $$Z_1$$ and $$A_2$$, $$Z_2$$, respectively. Then A and Z-sequence of the product $${\mathcal {R}}(g_1,f_1)*{\mathcal {R}}(g_2,f_2)$$ is equal to
\begin{aligned}&A(z)=A_2(z)\cdot A_1\left( \frac{z}{A_2(z)}\right) ,\\&Z(z)=\left( 1-\frac{z}{A_2(z)}Z_2(z)\right) \cdot Z_1(z)+A_1\left( \frac{z}{A_2(z)}\right) \cdot Z_2(z). \end{aligned}

Based on the above one can prove the following.

### Proposition 2.3

Let $${\mathcal {R}}(g_1,f_1,h_1)$$, $${\mathcal {R}}(g_2,f_2,h_2)$$ be 3-dimensional Riordan arrays with A, Z and H-sequences: $$A_1$$, $$Z_1$$, $$H_1$$ and $$A_2$$, $$Z_2$$, $$H_2$$, respectively. Then A, Z and H-sequence of the product $${\mathcal {R}}(g_1,f_1,h_1)*{\mathcal {R}}(g_2,f_2,h_2)$$ is equal to
\begin{aligned} A(z)= & {} A_2(z)\cdot A_1\left( \frac{z}{A_2(z)}\right) \\ Z(z)= & {} \left[ 1-\frac{z}{A_2(z)}Z_2(z)\right] \cdot Z_1\left( \frac{z}{A_2(z)}\right) +A_1\left( \frac{z}{A_2(z)}\right) \cdot Z_2(z)\\ H(z)= & {} H_2(z)\cdot H_1\left( \frac{z}{H_2(z)}\right) . \end{aligned}

### Proof

By Lemma 2.1, A and Z-sequence of $${\mathcal {R}}(g,f)$$ coincide with A and Z-sequence of $${\mathcal {R}}(g,f,h)$$, so two first equalities hold. By Theorem 1.1, h-sequence of $${\mathcal {R}}(g,f,h)$$ is equal to h, so from the definition and Lemma 2.1, we get that h-sequence of $${\mathcal {R}}(g_1(z)\cdot g_2(f_1(z)),f_2(f_1(z)),h_1(z)\cdot h_2(f_1(z)))$$ is the same as A-sequence of
\begin{aligned} {\mathcal {R}}(h_1(z)\cdot h_2(f_1(z)),f_2(f_1(z)))={\mathcal {R}}(h_1(z),f_1(z))*{\mathcal {R}}(h_2(z),f_2(z)). \end{aligned}
Thus, using Theorem 2.2 again, we get the third equality. $$\square$$

### 2.2 Possible Application

In this section we join the representation proposed in the first section with some other issue. Namely, the total positivity of a Riordan matrix. An infinite matrix is said to be totally positive (or shortly TP) if all its minors are nonnegative. In particular, a Toeplitz matrix
\begin{aligned} \begin{bmatrix} a_0&\quad&\quad&\quad&\\ a_1&\quad a_0&\quad&\quad&\\ a_2&\quad a_1&\quad a_0&\quad&\\ a_3&\quad a_2&\quad a_1&\quad a_0\quad&\\ \vdots&\quad&\quad&\quad&\quad \ddots \\ \end{bmatrix} \qquad \text {with all } a_n\ge 0, \end{aligned}
is totally positive if and only if $$a(z)=\sum _{n=0}^\infty a_nz^n$$ has only real (and nonpositive) zeros, and in this case $$(a_n)_{n=0}^\infty$$ is called Pólya frequency sequence.
Let’s get back to our matrices. It is obvious that fixing m in (1.3), one obtains a 2-dimensional Riordan array. It is called the m-th layer of $${\mathcal {R}}(g,f,h)$$. According to (1.3), the m-th layer of $${\mathcal {R}}(g,f,h)$$ is equal to $${\mathcal {R}}(gh^m,f)$$. From
\begin{aligned} {\mathcal {R}}(g(z)h^m(z),f(z))={\mathcal {R}}(h^m(z),z)*{\mathcal {R}}(g(z),f(z))=({\mathcal {R}}(h(z),z))^m*{\mathcal {R}}(g(z),f(z)), \end{aligned}
and the fact that the product of TP matrices is a TP matrix, we get the following conclusion.

### Corollary 2.4

If $${\mathcal {R}}(g(z),f(z))$$ and $${\mathcal {R}}(h(z),z)$$ are totally positive, then every layer of $${\mathcal {R}}(g(z),f(z),h(z))$$ is totally positive.

Totally positive matrices were considered in the context of A and Z sequences.

It was first proved in  (see also [10, 11]) that every 2D Riordan array $${\mathcal {R}}(g,f)$$ is induced by its production matrix
\begin{aligned} P_{{\mathcal {R}}(g,f)}=\begin{bmatrix} z_0&\quad a_0&\quad&\quad&\\ z_1&\quad a_1&\quad a_0&\quad&\\ z_2&\quad a_2&\quad a_1&\quad a_0&\\ \vdots&\quad&\quad&\quad&\quad \ddots \\ \end{bmatrix}. \end{aligned}
(2.1)
In particular, if we write U (as in ) for the shift matrix:
\begin{aligned} U=\begin{bmatrix} 0&\quad 1&\quad&\quad&\\ 0&\quad 0&\quad 1&\quad&\\ 0&\quad 0&\quad 0&\quad 1&\\ \vdots&\quad&\quad&\quad&\quad \ddots \\ \end{bmatrix}, \end{aligned}
then $$P_{{\mathcal {R}}(g,f)}$$ is the production matrix of the Riordan array $${\mathcal {R}}(g,f)$$ if and only if $$U{\mathcal {R}}(g,f)={\mathcal {R}}(g,f)P_{{\mathcal {R}}(g,f)}$$.

From  we know that if the production matrix $$P_{{\mathcal {R}}(g,f)}$$ is TP, then $${\mathcal {R}}(g,f)$$ is TP as well. Thus, we finish with the following observation.

### Corollary 2.5

If $$P_{{\mathcal {R}}(g,f)}$$ given by (2.1) is TP matrix and H is a Pólya frequency sequence, then every layer of $${\mathcal {R}}(g,f,h)$$ is a totally positive matrix.

### Example

It can be checked that for
\begin{aligned} A=(2,3,1,0,0,0,0,\ldots ), \qquad Z=(3,5,0,0,0,0,\ldots ) \end{aligned}
the production matrix $$P_{{\mathcal {R}}(g,f)}$$ is TP (see ). Moreover,
\begin{aligned} H=(2,5,4,1,0,0,0,0,\ldots ) \end{aligned}
is a Pólya frequency sequence. Thus, all the layers of $${\mathcal {R}}(g,f,h)$$
\begin{aligned} L_1= & {} \left[ {\begin{matrix} 1&{}&{}&{}&{}&{}&{}\\ 3&{}2&{}&{}&{}&{}&{}\\ 19&{}12&{}4&{}&{}&{}&{}\\ 117&{}78&{}36&{}8&{}&{}&{}\\ 741&{}504&{}272&{}96&{}16&{}&{}\\ 4743&{}3266&{}1830&{}848&{}240&{}32&{}\\ \vdots &{}&{}&{}&{}&{}&{}\ddots \\ \end{matrix}}\right] , \quad L_2=\left[ {\begin{matrix} 2&{}&{}&{}&{}&{}&{}\\ 11&{}4&{}&{}&{}&{}&{}\\ 57&{}34&{}8&{}&{}&{}&{}\\ 342&{}224&{}92&{}16&{}&{}&{}\\ 2146&{}1448&{}740&{}232&{}32&{}&{}\\ 13678&{}9376&{}5168&{}2208&{}560&{}64&{}\\ \vdots &{}&{}&{}&{}&{}&{}\ddots \\ \end{matrix}}\right] ,\\ L_3= & {} \left[ {\begin{matrix} 4&{}&{}&{}&{}&{}&{}\\ 32&{}8&{}&{}&{}&{}&{}\\ 177&{}88&{}16&{}&{}&{}&{}\\ 1015&{}634&{}224&{}32&{}&{}&{}\\ 6241&{}4156&{}1972&{}544&{}64&{}&{}\\ 39511&{}26922&{}14412&{}5640&{}1280&{}128&{}\\ \vdots &{}&{}&{}&{}&{}&{}\ddots \\ \end{matrix}}\right] , \quad L_4{=}\left[ {\begin{matrix} 8&{}&{}&{}&{}&{}&{}\\ 84&{}16&{}&{}&{}&{}&{}\\ 530&{}216&{}32&{}&{}&{}&{}\\ 3047&{}1740&{}528&{}64&{}&{}&{}\\ 18297&{}11842&{}5128&{}1248&{}128&{}&{}\\ 114464&{}77248&{}39596&{}14128&{}2880&{}256&{}\\ \vdots &{}&{}&{}&{}&{}&{}\ddots \\ \end{matrix}}\right] ,\\ L_5= & {} \left[ {\begin{matrix} 16&{}&{}&{}&{}&{}&{}\\ 208&{}32&{}&{}&{}&{}&{}\\ 1512&{}512&{}64&{}&{}&{}&{}\\ 9088&{}4624&{}1216&{}128&{}&{}&{}\\ 54033&{}33264&{}13024&{}2816&{}256&{}&{}\\ 333131&{}220882&{}106976&{}34752&{}6400&{}512&{}\\ \vdots &{}&{}&{}&{}&{}&{}\ddots \\ \end{matrix}}\right] , \,\, L_6{=}\left[ {\begin{matrix} 32&{}&{}&{}&{}&{}&{}\\ 496&{}64&{}&{}&{}&{}&{}\\ 4128&{}1184&{}128&{}&{}&{}&{}\\ 26584&{}11936&{}2752&{}256&{}&{}&{}\\ 159762&{}91728&{}32384&{}6272&{}512&{}&{}\\ 974291&{}627092&{}284000&{}84096&{}14080&{}1024&{}\\ \vdots &{}&{}&{}&{}&{}&{}\ddots \\ \end{matrix}}\right] , \cdots \end{aligned}
are totally positive.

Let’s finish this short note with some remarks about possible generalizations of the presented notions. In  the authors proposed extending the definition of 2-dimensional Riordan array given by
\begin{aligned} r_{nk}=[z^n]gf^k\qquad n,k\in \mathbf {N}_0 \end{aligned}
to all $$n,k\in \mathbb {Z}$$, and called them recursive matrices (in  they are also called complementary). Also in 3-dimensional case one can introduce 3-dimensional recursive matrix $${}_{\mathbb {Z}} R=[r_{nkm}]$$ whose entries are given by (1.3) for all $$n,k,m\in \mathbb {Z}$$. It is interesting that, by  (see Section 3 of this paper) for such $${}_{\mathbb {Z}} R$$ the following identities hold:
\begin{aligned} r_{n+m,k+m,p}= & {} \sum _{j=0}^{n-k}a_j^{(m)}r_{n,k+j,p},\\ r_{n+m,k+m,p}= & {} \sum _{j=0}^{n-k}f_{j+m}^{(m)}r_{n-j,k,p}, \end{aligned}
where by $$a_j^{(m)}$$ we mean $$[z^j]A^m$$.

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