Multiple Almost-Riordan Arrays

• Multiple almost-Riordan arrays are lower-triangular arrays of formal power series defined by a decoupled first column and a cyclically generated Riordan tail.
• They exhibit a rich group structure with specialized A-, Z-, and W-sequence characterizations that generalize classical Riordan array theory.
• A novel compression operation transforms the periodic ladder into a reparameterized sequence framework, preserving key production matrix and recurrence relations.

Searching arXiv for the cited papers on multiple almost-Riordan arrays and related work. Multiple almost-Riordan arrays are lower-triangular arrays of formal-power-series type that extend both almost-Riordan arrays and multiple Riordan arrays by decoupling the first column from an otherwise cyclically generated Riordan tail. In the formulation developed in "Sequence Characterization of Multiple Almost-Riordan Arrays and Their Compressions" (He, 2 Sep 2025), a multiple almost-Riordan array (MARA) is determined by a first-column series $b(t)$, a tail prefactor $g(t)$, and a finite suite of multipliers $f_1,\ldots,f_\ell$, with $\ell \ge 2$, subject to congruence-class restrictions $b,g\in \mathbb K[[t^\ell]]$ and $f_j\in t\mathbb K[[t^\ell]]$. The theory places MARAs in a group, furnishes sequence and production-matrix characterizations, and defines a compression operation preserving a transformed sequence structure (He, 2 Sep 2025). The $\ell=2$ case, treated in detail as double almost-Riordan arrays, provides the prototype from which the general multiple theory is organized (He, 2024).

1. Definition and placement within Riordan theory

The classical Riordan array is a lower-triangular matrix associated with a pair $(g(t),f(t))$, where $g$ has order $0$ and $g(t)$0 has order $g(t)$1, with entries

$g(t)$2

Its group law is

$g(t)$3

with identity $g(t)$4 (He, 2 Sep 2025). An almost-Riordan array replaces the first column by an independent series: $g(t)$5 and its multiplication law is

$g(t)$6

(He, 2 Sep 2025); see also the foundational single-variable treatment of almost-Riordan arrays (Barry, 2016).

A multiple Riordan array introduces $g(t)$7 cyclic multipliers. With $g(t)$8 and $g(t)$9, its columns are

$f_1,\ldots,f_\ell$0

and the entry formula is

$f_1,\ldots,f_\ell$1

(He, 2 Sep 2025).

The MARA combines these two extensions. Definition 6.1 of (He, 2 Sep 2025) sets

$f_1,\ldots,f_\ell$2

with entries

$f_1,\ldots,f_\ell$3

and, for $f_1,\ldots,f_\ell$4,

$f_1,\ldots,f_\ell$5

Admissibility requires $f_1,\ldots,f_\ell$6, $f_1,\ldots,f_\ell$7, and $f_1,\ldots,f_\ell$8, so that

$f_1,\ldots,f_\ell$9

has a compositional inverse (He, 2 Sep 2025).

Several specializations are explicit. Setting $\ell \ge 2$0 recovers classical almost-Riordan arrays; setting $\ell \ge 2$1 and $\ell \ge 2$2 for all $\ell \ge 2$3 gives the Appell-type Riordan array $\ell \ge 2$4; setting $\ell \ge 2$5 leaves a quasi-Riordan-like initial column attached to a multiple Riordan tail (He, 2 Sep 2025). The double theory of (He, 2024) is the $\ell \ge 2$6 specialization, with parity constraints $\ell \ge 2$7 even and $\ell \ge 2$8 odd.

2. Group structure and fundamental action

The central algebraic result is that MARAs form a group, called the multiple almost-Riordan group (He, 2 Sep 2025). If

$\ell \ge 2$9

then the multiplication law is

$b,g\in \mathbb K[[t^\ell]]$0

and the inverse is

$b,g\in \mathbb K[[t^\ell]]$1

where $b,g\in \mathbb K[[t^\ell]]$2 is the compositional inverse of $b,g\in \mathbb K[[t^\ell]]$3 (He, 2 Sep 2025). The identity is

$b,g\in \mathbb K[[t^\ell]]$4

The group law is inherited from a fundamental theorem describing the action of a MARA on series supported on congruence classes modulo $b,g\in \mathbb K[[t^\ell]]$5. The paper states the output $b,g\in \mathbb K[[t^\ell]]$6 for

$b,g\in \mathbb K[[t^\ell]]$7

in several cases: $b,g\in \mathbb K[[t^\ell]]$8

$b,g\in \mathbb K[[t^\ell]]$9

$f_j\in t\mathbb K[[t^\ell]]$0

(He, 2 Sep 2025). In the double case this action splits according to parity of the input series and is expressed through

$f_j\in t\mathbb K[[t^\ell]]$1

which homogenizes the alternating ladder (He, 2024). The multiple formulation replaces parity splitting by a congruence-class decomposition modulo $f_j\in t\mathbb K[[t^\ell]]$2.

This construction is structurally distinct from the earlier hierarchy $f_j\in t\mathbb K[[t^\ell]]$3 studied in the single-tail setting, where successive additional leading columns are prepended to an almost-Riordan array (Barry, 2016). A plausible implication is that the later MARA theory of (He, 2 Sep 2025) should be viewed not as that hierarchy itself, but as a cyclic-multiplier generalization based on the multiple Riordan group.

3. Sequence characterization

The sequence characterization of MARAs generalizes the $f_j\in t\mathbb K[[t^\ell]]$4- and $f_j\in t\mathbb K[[t^\ell]]$5-sequence formalism of Riordan theory. The main MARA theorem in (He, 2 Sep 2025) adopts a decomposition

$f_j\in t\mathbb K[[t^\ell]]$6

where $f_j\in t\mathbb K[[t^\ell]]$7 encodes the first column and the trimmed blocks $f_j\in t\mathbb K[[t^\ell]]$8 are Riordan arrays with common core

$f_j\in t\mathbb K[[t^\ell]]$9

From this one obtains one $\ell=2$0-sequence, $\ell=2$1 $\ell=2$2-sequences, and one $\ell=2$3-sequence.

Writing

$\ell=2$4

Theorem new-3.1 gives

$\ell=2$5

$\ell=2$6

$\ell=2$7

$\ell=2$8

and

$\ell=2$9

(He, 2 Sep 2025).

Two aspects are notable. First, the single $(g(t),f(t))$0-sequence governs the $(g(t),f(t))$1-periodic tail, just as the classical $(g(t),f(t))$2-sequence governs the shift structure of a Riordan array. Second, the decoupled first column affects the $(g(t),f(t))$3- and $(g(t),f(t))$4-components. This is the multiple analog of the extra first-column data in the almost-Riordan setting described in (Barry, 2016).

The paper also reports an alternative viewpoint, inherited from the theory of multiple Riordan arrays, in which one uses $(g(t),f(t))$5 $(g(t),f(t))$6-sequences together with a $(g(t),f(t))$7-sequence and a $(g(t),f(t))$8-sequence; however, the principal MARA characterization in (He, 2 Sep 2025) adopts the one-$(g(t),f(t))$9, $g$0-$g$1, one-$g$2 scheme. In the double case, this specializes to one $g$3-sequence, two $g$4-sequences, and one $g$5-sequence (He, 2024).

4. Production matrices and row generation

The production matrix encodes the row-by-row generation of the array. For MARAs, Theorem 4.3-2 states that the production matrix $g$6 has column generating functions

$g$7

(He, 2 Sep 2025). This extends the classical Riordan form $g$8 by inserting the extra columns required for the first-column dynamics and the $g$9-periodic tail (He, 2 Sep 2025).

The paper explains that, as in the Riordan case,

$0$0

with $0$1 the upper shift and $0$2 the truncated MARA, although in practice the analysis proceeds directly from the listed column generating functions (He, 2 Sep 2025). In the earlier almost-Riordan theory, the production matrix likewise has a distinguished first column, a $0$3-column, and then the $0$4-columns of the Riordan tail (Barry, 2016). The MARA formula should therefore be read as the cyclic extension of that pattern.

The $0$5 example in (He, 2 Sep 2025) uses

$0$6

for which

$0$7

$0$8

$0$9

Hence

$g(t)$00

(He, 2 Sep 2025).

The $g(t)$01 example matches the double almost-Riordan construction: $g(t)$02 with

$g(t)$03

and

$g(t)$04

(He, 2 Sep 2025). The same formulas are obtained in the dedicated double paper (He, 2024).

5. Compression and transformed sequence structure

Compression is a central operation in the multiple theory. For a MARA $g(t)$05, its compression is defined by

$g(t)$06

(He, 2 Sep 2025). This generalizes the double compression

$g(t)$07

from (He, 2024).

The compressed parameter series are given in (He, 2 Sep 2025) by

$g(t)$08

The theorem then states

$g(t)$09

and for $g(t)$10,

$g(t)$11

for residues $g(t)$12 (He, 2 Sep 2025).

Compression preserves a transformed sequence characterization. Theorem 9.3 expresses the compressed relations in terms of the original $g(t)$13-, $g(t)$14-, and $g(t)$15-sequences. In generating-function form,

$g(t)$16

$g(t)$17

$g(t)$18

$g(t)$19

$g(t)$20

(He, 2 Sep 2025).

The same theorem gives linear recurrences for compressed entries. This suggests that compression converts the original $g(t)$21-periodic ladder into a single transformed ladder while preserving the same sequence data in reparameterized form. In the double case, (He, 2024) states the corresponding formulas with $g(t)$22.

6. Examples, related structures, and scope of current knowledge

Two worked families illustrate the framework. The first is the double example

$g(t)$23

whose initial rows, production matrix, and compressed array are computed explicitly in (He, 2 Sep 2025). Its compression begins

$g(t)$24

and is connected to the Fibonacci–Stanley tree (He, 2 Sep 2025). The same construction appears in the double paper, where the corresponding double Riordan compression is identified with the Fibonacci–Stanley array (He, 2024).

The second is the $g(t)$25 family

$g(t)$26

whose displayed matrix, sequences, and compression are given explicitly in (He, 2 Sep 2025). The compressed array is

$g(t)$27

with the compressed sequences verified in Remark 4.1 of that paper.

The group also admits a nontrivial subgroup structure. For normalized MARAs with $g(t)$28, (He, 2 Sep 2025) identifies an Appell subgroup $g(t)$29, a Lagrange subgroup $g(t)$30, and type-$g(t)$31 Bell subgroups $g(t)$32. The Appell subgroup is normal, and the paper states a semidirect product decomposition

$g(t)$33

It also records derivative and lifted Bell-type subgroups, thereby extending subgroup patterns already familiar from Riordan and almost-Riordan theory (He, 2 Sep 2025).

A common misconception is to identify MARAs with arbitrary arrays having several free initial columns. The papers do not make that definition. In (He, 2 Sep 2025), the term refers specifically to the cyclic-multiplier structure

$g(t)$34

with one free first column and a tail generated by $g(t)$35 and the ordered suite $g(t)$36. By contrast, (Barry, 2016) studies an earlier hierarchy of iterated almost-Riordan constructions $g(t)$37, where additional leading columns are prepended before a single Riordan tail. These are related generalizations, but they are not the same formal object.

The present research trajectory is therefore layered. The single almost-Riordan group establishes the first-column extension of Riordan theory (Barry, 2016). The double theory proves the two-multiplier case, including total positivity for selected classes and a compression formalism (He, 2024). The general multiple theory then systematizes the $g(t)$38-multiplier case, providing the group law, sequence characterization, production matrices, compression, examples, and subgroup structure (He, 2 Sep 2025). The papers further point to open directions involving total positivity, improper arrays, LMMS-type subgroups, interaction between compression and group operations, $g(t)$39-analogs, and higher-dimensional extensions (He, 2 Sep 2025).