Sequence Characterization of 3-Dimensional Riordan Arrays and Some Application

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.


Introduction
Let's recall that the Riordan group, introduced in [1], is a group of N 0 × N 0 matrices that are identified with pairs of formal power series. Namely, denoting by F 0 -the ring of formal power series with nonzero free term, and by F 1 -the ring of formal power series with zero free term but nonzero the next term, the Riordan group R consists of pairs R(g, f ) with g ∈ F 0 , f ∈ F 1 . The multiplication of these pairs is given by R(g 1 (z), f 1 (z)) * R(g 2 (z), f 2 (z)) = R(g 1 (z) · g 2 (f 1 (z)), f 2 (f 1 (z))), 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)- , is defined by the formula: (1.1) In this note we are interested in R 3 -the group of 3-dimensional Riordan arrays. It was proved by Cheon and Jin [5] that R 3 is an extension of R by F 0 . In this group the matrices R = [r nkm ] n,k,m∈N0 are associated with the triples of series (g, f, h) with g, h ∈ F 0 , f ∈ F 1 . The multiplication of such triples is defined as follows: Each entry of R = [r nkm ] can be found from the relation: 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: (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: (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 After discussing the above presentation, we will propose how it can be used to obtain some totally positive Riordan arrays.

Sequence Characterization
To prove the main result, it suffices to notice that the below lemma holds.
Lemma 2.1. The groups Proof. Clearly, the maps φ g : establish the desired isomorphism, and the correspondence of sequences.
For 2-dimensional Riordan arrays the following result was obtained by He and Sprugnoli.
Based on the above one can prove the following. .
Thus, using Theorem 2.2 again, we get the third equality.

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 T P ) if all its minors are nonnegative.
In particular, a Toeplitz matrix ⎡ with all a n ≥ 0, is totally positive if and only if a(z) = ∞ n=0 a n z n has only real (and nonpositive) zeros, and in this case (a n ) ∞ n=0 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 R(g, f, h). According to (1.3), the m-th layer of R(g, f, h) is equal to R(gh m , f). From R(g(z)h m (z), f(z)) = R(h m (z), z) * R(g(z), f(z)) = (R(h(z), z)) m * R(g(z), f(z)), and the fact that the product of T P matrices is a T P matrix, we get the following conclusion. Totally positive matrices were considered in the context of A and Z sequences.
It was first proved in [9] (see also [10,11]) that every 2D Riordan array R(g, f ) is induced by its production matrix In particular, if we write U (as in [11]) for the shift matrix: then P R(g,f ) is the production matrix of the Riordan array R(g, f ) if and only if U R(g, f ) = R(g, f )P R(g,f ) .
From [12] we know that if the production matrix P R(g,f ) is T P , then R(g, f ) is T P as well. Thus, we finish with the following observation. the production matrix P R(g,f ) is T P (see [12]