---
title: Multiple Almost-Riordan Arrays
url: https://www.emergentmind.com/topics/multiple-almost-riordan-arrays
type: topic
---

# Multiple Almost-Riordan Arrays

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" [2509.02893], 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 [2509.02893]. The \(\ell=2\) case, treated in detail as double almost-Riordan arrays, provides the prototype from which the general multiple theory is organized [2411.05334].

## 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 \(f\) has order \(1\), with entries
\[
d_{n,k}=[t^n]\,g(t)f(t)^k.
\]
Its group law is
\[
(g_1,f_1)(g_2,f_2)=\big(g_1\,g_2(f_1),\,f_2(f_1)\big),
\]
with identity \((1,t)\) [2509.02893]. An almost-Riordan array replaces the first column by an independent series:
\[
(a|\,g,f)=(a,\;tg,\;tgf,\;tgf^2,\ldots),
\]
and its multiplication law is
\[
(a|\,g,f)(b|\,d,h)=\left(\left. a+\frac{tg}{f}\big(b(f)-1\big)\right|\,g\,d(f),\,h(f)\right)
\]
[2509.02893]; see also the foundational single-variable treatment of almost-Riordan arrays [1606.05077].

A multiple Riordan array introduces \(\ell\) cyclic multipliers. With \(g\in \mathbb K[[t^\ell]]\) and \(f_j\in t\mathbb K[[t^\ell]]\), its columns are
\[
(g,\;g f_1,\;g f_1f_2,\;\ldots,\;g f_1f_2\cdots f_\ell,\;g f_1^2f_2\cdots f_\ell,\;\ldots),
\]
and the entry formula is
\[
d_{n,k} = [t^n]\; g(t)\, f_1(t)^{\left\lfloor \frac{k+\ell-1}{\ell}\right\rfloor}
f_2(t)^{\left\lfloor \frac{k+\ell-2}{\ell}\right\rfloor}\cdots
f_\ell(t)^{\left\lfloor \frac{k}{\ell}\right\rfloor}
\]
[2509.02893].

The MARA combines these two extensions. Definition 6.1 of [2509.02893] sets
\[
(b|g;f_1,\ldots,f_\ell)
=
\big(b,\;tg,\;tgf_1,\;tgf_1f_2,\;\ldots,\;tgf_1f_2\cdots f_\ell,\;tg\,f_1^2f_2\cdots f_\ell,\;\ldots\big),
\]
with entries
\[
d_{n,0}=[t^n]\, b(t),
\]
and, for \(k\ge 1\),
\[
d_{n,k}=[t^{n-1}]\, g(t)\, f_1(t)^{\left\lfloor \frac{k+\ell -2}{\ell}\right\rfloor}
f_2(t)^{\left\lfloor \frac{k+\ell -3}{\ell}\right\rfloor}
\cdots
f_\ell(t)^{\left\lfloor \frac{k-1}{\ell}\right\rfloor}.
\]
Admissibility requires \(b_0\neq 0\), \(g_0\neq 0\), and \(f_{j,1}\neq 0\), so that
\[
h=\sqrt[\ell]{f_1\cdots f_\ell}
\]
has a compositional inverse [2509.02893].

Several specializations are explicit. Setting \(\ell=1\) recovers classical almost-Riordan arrays; setting \(b=g\) and \(f_j=t\) for all \(j\) gives the Appell-type Riordan array \((g,t)\); setting \(b=1\) leaves a quasi-Riordan-like initial column attached to a multiple Riordan tail [2509.02893]. The double theory of [2411.05334] is the \(\ell=2\) specialization, with parity constraints \(b,g\) even and \(f_1,f_2\) odd.

## 2. Group structure and fundamental action

The central algebraic result is that MARAs form a group, called the multiple almost-Riordan group [2509.02893]. If
\[
h=\sqrt[\ell]{f_1\cdots f_\ell},
\]
then the multiplication law is
\[
(b|g;f_1,\ldots,f_\ell)(c|d;h_1,\ldots,h_\ell)
=
\left(\left.
c_0\, b + \frac{t g}{f_\ell}\big(c(h)-c_0\big)
\right|\, g\, d(h),\; \frac{f_1}{h}h_1(h),\; \ldots,\; \frac{f_\ell}{h}h_\ell(h)\right),
\]
and the inverse is
\[
(b|g;f_1,\ldots,f_\ell)^{-1}
=
\left(\left.
\frac{1}{b_0} + \frac{f_\ell(\overline{h})}{b_0\, \overline{h}\, g(\overline{h})}\big(b_0-b(\overline{h})\big)
\right|
\frac{1}{g(\overline{h})};\;
\overline{h}\frac{t}{f_1(\overline{h})},\; \ldots,\;
\overline{h}\frac{t}{f_\ell (\overline{h})}\right),
\]
where \(\overline h\) is the compositional inverse of \(h\) [2509.02893]. The identity is
\[
(1|\,1;\,t,\ldots,t).
\]

The group law is inherited from a fundamental theorem describing the action of a MARA on series supported on congruence classes modulo \(\ell\). The paper states the output \(v(t)\) for
\[
(b|g;f_1,\ldots,f_\ell)u(t)=v(t)
\]
in several cases:
\[
v(t)=u_0\, b(t)+\frac{t g(t)}{f_\ell(t)}\big(u(h)-u_0\big),
\]
\[
v(t)=\frac{t g(t)}{h(t)}\,u(h(t)),
\]
\[
v(t)=\frac{t g(t)\, f_1(t)\cdots f_{j-1}(t)}{h(t)^j}\, u(h(t))
\qquad (j=2,\ldots,\ell)
\]
[2509.02893]. In the double case this action splits according to parity of the input series and is expressed through
\[
\phi(t)=\sqrt{f_1(t)f_2(t)},
\]
which homogenizes the alternating ladder [2411.05334]. The multiple formulation replaces parity splitting by a congruence-class decomposition modulo \(\ell\).

This construction is structurally distinct from the earlier hierarchy \(R(2),R(3),\ldots\) studied in the single-tail setting, where successive additional leading columns are prepended to an almost-Riordan array [1606.05077]. A plausible implication is that the later MARA theory of [2509.02893] 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 \(A\)- and \(Z\)-sequence formalism of Riordan theory. The main MARA theorem in [2509.02893] adopts a decomposition
\[
D=D_0+D_1+\cdots+D_\ell,
\]
where \(D_0\) encodes the first column and the trimmed blocks \(D_j^*\) are Riordan arrays with common core
\[
h=\sqrt[\ell]{f_1f_2\cdots f_\ell}.
\]
From this one obtains one \(A\)-sequence, \(\ell\) \(Z\)-sequences, and one \(W\)-sequence.

Writing
\[
A(t)=\sum_{k\ge 0}a_k t^{\ell k},\qquad
Z_j(t)=\sum_{k\ge 0} z_{j,k} t^{\ell k}\quad(j=1,\ldots,\ell),\qquad
W(t)=\sum_{k\ge 0} w_k t^{\ell k},
\]
Theorem new-3.1 gives
\[
A(t)=\frac{t^\ell}{\overline{h}^\ell},
\]
\[
Z_1(t)=\frac{1}{\overline{h}^\ell}\left(1-\frac{g_0}{g(\overline{h})}\right),
\]
\[
Z_m(t)=
\dfrac{1}{\overline{h}^\ell}\left(
1-\dfrac{g_0 f_{1,1} f_{2,1}\cdots f_{m-1,1}\,\overline{h}^{m-1}}
{g(\overline{h})\, f_1(\overline{h}) f_2(\overline{h})\cdots f_{m-1}(\overline{h})}
\right),
\qquad m=2,3,\ldots,\ell-1,
\]
\[
Z_\ell(t)=
\frac{g_0 f_{1,1} f_{2,1}\cdots f_{\ell-1,1}}{b_0}
+\frac{t^\ell}{\overline{h}^\ell}
-\frac{g_0 f_{1,1} f_{2,1}\cdots f_{\ell -1,1}\, b(\overline{h})\, f_\ell(\overline{h})}
{b_0\, \overline{h}\, g(\overline{h})},
\]
and
\[
W(t)=
\frac{f_\ell(\overline{h})\big((1-w_0 \overline{h}^\ell) b(\overline{h})-b_0\big)}
{\overline{h}^{\ell+1} g(\overline{h})}+w_0,
\qquad
w_0=\frac{b_\ell}{b_0}
\]
[2509.02893].

Two aspects are notable. First, the single \(A\)-sequence governs the \(\ell\)-periodic tail, just as the classical \(A\)-sequence governs the shift structure of a Riordan array. Second, the decoupled first column affects the \(Z_\ell\)- and \(W\)-components. This is the multiple analog of the extra first-column data in the almost-Riordan setting described in [1606.05077].

The paper also reports an alternative viewpoint, inherited from the theory of multiple Riordan arrays, in which one uses \(\ell\) \(A\)-sequences together with a \(Z\)-sequence and a \(W\)-sequence; however, the principal MARA characterization in [2509.02893] adopts the one-\(A\), \(\ell\)-\(Z\), one-\(W\) scheme. In the double case, this specializes to one \(A\)-sequence, two \(Z\)-sequences, and one \(W\)-sequence [2411.05334].

## 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 \(P\) has column generating functions
\[
P=\big(W(t),\; tZ_1(t),\; \ldots,\; t^{\ell-1}Z_{\ell -1}(t),\; Z_\ell(t),\; tA(t),\; t^2A(t),\; t^3A(t),\; \ldots \big)
\]
[2509.02893]. This extends the classical Riordan form \(P=(Z(t),A(t),tA(t),\ldots)\) by inserting the extra columns required for the first-column dynamics and the \(\ell\)-periodic tail [2509.02893].

The paper explains that, as in the Riordan case,
\[
P = R^{-1} U R,
\]
with \(U\) the upper shift and \(R\) the truncated MARA, although in practice the analysis proceeds directly from the listed column generating functions [2509.02893]. In the earlier almost-Riordan theory, the production matrix likewise has a distinguished first column, a \(Z\)-column, and then the \(A\)-columns of the Riordan tail [1606.05077]. The MARA formula should therefore be read as the cyclic extension of that pattern.

The \(\ell=3\) example in [2509.02893] uses
\[
\left(\frac{1}{1-t^6}\;\middle|\;\frac{1}{1-t^3};\ \frac{t}{1-t^3},\ t(1+t^3),\ \frac{t}{1+t^3}\right),
\]
for which
\[
A(t)=1+t^3,\qquad Z_1(t)=1,\qquad Z_2(t)=1+\frac{1}{1+t^3}=1-t^3+t^6-\cdots,
\]
\[
Z_3(t)=2+t^3-\frac{(1+t^3)^2}{(1+2t^3)^2}=1+3t^3-5t^6+12t^9-\cdots,
\]
\[
W(t)=\frac{t^3(1+t^3)}{(1+2t^3)^2}=t^3-3t^6+8t^9-\cdots.
\]
Hence
\[
P=\big(W(t),\ tZ_1(t),\ t^2Z_2(t),\ Z_3(t),\ tA(t),\ t^2A(t),\ldots\big)
\]
[2509.02893].

The \(\ell=2\) example matches the double almost-Riordan construction:
\[
\left(\frac{1}{1-t^4}\;\middle|\;\frac{1}{1-t^2};\ t,\ \frac{t}{1-t^2}\right),
\]
with
\[
A(t)=1+t^2,\qquad Z_1(t)=1,\qquad
Z_2(t)=\frac{1+3t^2+t^4}{1+2t^2},\qquad
W(t)=\frac{t^2(1+t^2)}{1+2t^2},
\]
and
\[
P=(W(t),\, tZ_1(t),\, Z_2(t),\, tA(t),\, t^2A(t),\ldots)
\]
[2509.02893]. The same formulas are obtained in the dedicated double paper [2411.05334].

## 5. Compression and transformed sequence structure

Compression is a central operation in the multiple theory. For a MARA \(D=(d_{n,k})_{n\ge k\ge 0}\), its compression is defined by
\[
\hat d_{n,k}:=d_{\ell n-(\ell-1)k,\;k},\qquad n\ge k\ge 0
\]
[2509.02893]. This generalizes the double compression
\[
\hat d_{n,k}=d_{2n-k,\;k}
\]
from [2411.05334].

The compressed parameter series are given in [2509.02893] by
\[
\hat b(t)=\sum_{k\ge 0} b_{2k}t^k,\qquad
\hat g(t)=\sum_{k\ge 0} g_{2k}t^k,\qquad
\hat f_j(t)=\sum_{k\ge 0} f_{j,2k+1} t^{k+1}\quad(j=1,\ldots,\ell).
\]
The theorem then states
\[
\hat d_{n,0}=[t^n]\hat b(t),
\]
and for \(k\ge 1\),
\[
\hat d_{n,k}= \begin{cases}
[t^n]\; t\, \hat g\; \big(\hat f_1\hat f_2\cdots \hat f_\ell\big)^{\frac{k-1}{\ell}}, & k \equiv 1\ (\mathrm{mod}\ \ell),\\[0.6em]
[t^n]\; t\, \hat g\, \hat f_1\cdots \hat f_{m-1}\; \big(\hat f_1\hat f_2\cdots \hat f_\ell\big)^{\frac{k-m}{\ell}}, & k \equiv m\ (\mathrm{mod}\ \ell),
\end{cases}
\]
for residues \(m=2,\ldots,\ell\) [2509.02893].

Compression preserves a transformed sequence characterization. Theorem 9.3 expresses the compressed relations in terms of the original \(A\)-, \(Z_j\)-, and \(W\)-sequences. In generating-function form,
\[
A\!\left(\sqrt[\ell]{\frac{\hat f_1\hat f_2\cdots \hat f_\ell}{t^{\ell-1}}}\right)
=
\frac{\hat f_1\hat f_2\cdots \hat f_\ell}{t^\ell},
\]
\[
Z_1\!\left(\sqrt[\ell]{\frac{\hat f_1\hat f_2\cdots \hat f_\ell}{t^{\ell-1}}}\right)
=
\frac{1}{t}\left(1-\frac{g_0}{\hat g}\right),
\]
\[
Z_m\!\left(\sqrt[\ell]{\frac{\hat f_1\hat f_2\cdots \hat f_\ell}{t^{\ell-1}}}\right)
=
\frac{1}{t}\left(1-\frac{g_0 f_{1,1}f_{2,1}\cdots f_{m-1,1}\, t^{m-1}}{\hat g\, \hat f_1\hat f_2\cdots \hat f_m}\right),
\qquad m=2,\ldots,\ell-1,
\]
\[
Z_\ell\!\left(\sqrt[\ell]{\frac{\hat f_1\hat f_2\cdots \hat f_\ell}{t^{\ell-1}}}\right)
=
\frac{\hat f_\ell}{t^\ell \hat g}\big(\hat g \hat f_1\cdots \hat f_{\ell-1}-z_{\ell,0} t^{\ell-1}\hat b\big)+z_{\ell,0},
\]
\[
W\!\left(\sqrt[\ell]{\frac{\hat f_1\hat f_2\cdots \hat f_\ell}{t^{\ell-1}}}\right)
=
\frac{\hat f_\ell}{t^2 \hat g}\big(\hat b(1-w_0 t)-b_0\big)+w_0
\]
[2509.02893].

The same theorem gives linear recurrences for compressed entries. This suggests that compression converts the original \(\ell\)-periodic ladder into a single transformed ladder while preserving the same sequence data in reparameterized form. In the double case, [2411.05334] states the corresponding formulas with \(\psi(t)=\sqrt{\hat f_1\hat f_2/t}\).

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

Two worked families illustrate the framework. The first is the double example
\[
\left(\frac{1}{1-t^4}\;\middle|\;\frac{1}{1-t^2};\ t,\ \frac{t}{1-t^2}\right),
\]
whose initial rows, production matrix, and compressed array are computed explicitly in [2509.02893]. Its compression begins
\[
\begin{bmatrix}
1&0&0&0&0&0&0&0&0&\cdots\\
0&1&0&0&0&0&0&0&0&\cdots\\
1&1&1&0&0&0&0&0&0&\cdots\\
0&1&1&1&0&0&0&0&0&\cdots\\
1&1&1&2&1&0&0&0&0&\cdots\\
0&1&1&3&2&1&0&0&0&\cdots\\
1&1&1&4&3&3&1&0&0&\cdots\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots
\end{bmatrix}
\]
and is connected to the Fibonacci–Stanley tree [2509.02893]. The same construction appears in the double paper, where the corresponding double Riordan compression is identified with the Fibonacci–Stanley array [2411.05334].

The second is the \(\ell=3\) family
\[
\left(\frac{1}{1-t^6}\;\middle|\;\frac{1}{1-t^3};\ \frac{t}{1-t^3},\ t(1+t^3),\ \frac{t}{1+t^3}\right),
\]
whose displayed matrix, sequences, and compression are given explicitly in [2509.02893]. The compressed array is
\[
\left(\frac{1}{1-t^2}\;\middle|\;\frac{1}{1-t};\ \frac{t}{1-t},\ t(1+t),\ \frac{t}{1+t}\right),
\]
with the compressed sequences verified in Remark 4.1 of that paper.

The group also admits a nontrivial subgroup structure. For normalized MARAs with \(b(0)=g(0)=1\), [2509.02893] identifies an Appell subgroup \(\{(b|g;t,\ldots,t)\}\), a Lagrange subgroup \(\{(1|1;f_1,\ldots,f_\ell)\}\), and type-\(j\) Bell subgroups \(\{(b|g;f_1,\ldots,f_\ell): f_j=tg\}\). The Appell subgroup is normal, and the paper states a semidirect product decomposition
\[
{\cal M}a{\cal R}=\mathcal{A}_{\mathrm{Appell}\rtimes \mathcal{L}_{\mathrm{Lagrange}}.
\]
It also records derivative and lifted Bell-type subgroups, thereby extending subgroup patterns already familiar from Riordan and almost-Riordan theory [2509.02893].

A common misconception is to identify MARAs with arbitrary arrays having several free initial columns. The papers do not make that definition. In [2509.02893], the term refers specifically to the cyclic-multiplier structure
\[
(b|g;f_1,\ldots,f_\ell),
\]
with one free first column and a tail generated by \(g\) and the ordered suite \(f_1,\ldots,f_\ell\). By contrast, [1606.05077] studies an earlier hierarchy of iterated almost-Riordan constructions \(R(2),R(3),\ldots\), 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 [1606.05077]. The double theory proves the two-multiplier case, including total positivity for selected classes and a compression formalism [2411.05334]. The general multiple theory then systematizes the \(\ell\)-multiplier case, providing the group law, sequence characterization, production matrices, compression, examples, and subgroup structure [2509.02893]. The papers further point to open directions involving total positivity, improper arrays, LMMS-type subgroups, interaction between compression and group operations, \(q\)-analogs, and higher-dimensional extensions [2509.02893].

Source: https://www.emergentmind.com/topics/multiple-almost-riordan-arrays