The Double Almost-Riordan Arrays and Their Sequence Characterization, Compression, and Total Positivity

TL;DR

This work extends the Riordan framework by introducing double almost-Riordan arrays and the associated double almost-Riordan group, defined via elements $(b|g; f_1,f_2)$ with an explicit multiplication and two first fundamental theorem forms. It provides a detailed sequence characterization in terms of $A_1$/$A_2$-sequences, $Z$-sequences, and a $W$-sequence, along with explicit generating-function relations and a production-matrix description that underpins the ECO-style growth rules. The paper then develops a compression theory to produce compressed double Riordan and compressed double almost-Riordan arrays, giving exact formulas for compressed entries and showing how they relate to the Fibonacci-Stanley construction. A central contribution is the study of total positivity: it gives criteria under which compressed arrays are totally positive, explains how TP can be preserved or achieved via suitable choice of the $b$-block, and presents both positive results and counterexamples to illustrate the limits of TP transfer across decompositions.

Abstract

In this paper, we define double almost-Riordan arrays and find that the set of all double almost-Riordan arrays forms a group, called the double almost-Riordan group. We also obtain the sequence characteristics of double almost-Riordan arrays and give the production matrices for double almost-Riordan arrays. We define the compression of double almost-Riordan arrays and present their sequence characterization. Finally we give a characteristic for the total positivity of double Riordan arrays, by using which we discuss the total positivity for compressed double almost-Riordan arrays.

Figures (1)

• Figure 1: Figure $1$. Fibonacci tree $F_1$

Theorems & Definitions (33)

• Proposition 1.1
• Theorem 1.2
• Definition 1.3
• Theorem 1.4
• Definition 1.5
• Theorem 1.6
• Theorem 1.7
• Definition 2.1
• Theorem 2.2
• proof