Multiple Almost-Riordan Group

• Multiple almost-Riordan group is a set of lower-triangular infinite matrices with a distinguished first column and ℓ-periodic pattern.
• It generalizes classical and double almost-Riordan arrays by integrating a structured sequence characterization using A-, Z-, and W-data.
• The framework provides explicit group laws, inversion formulas, and a compression mechanism that aids combinatorial and algebraic analysis.

The multiple almost-Riordan group is a group of lower-triangular infinite matrices obtained by adjoining a distinguished first column to a multiple Riordan array. In the formulation introduced by He, an element has the form $(b|g;f_1,\dots,f_\ell)$, where $b$ and $g$ are series in $\mathbb K[[t^\ell]]$ and each $f_j$ lies in $t\mathbb K[[t^\ell]]$; its columns are generated by the pattern $(b,\; tg,\; tgf_1,\; tgf_1f_2,\dots)$, and the set of all such arrays forms a group under a composition law induced by a residue-class version of the first fundamental theorem for Riordan-type arrays (He, 2 Sep 2025). The construction extends the classical almost-Riordan group and the multiple Riordan group, while the double almost-Riordan group is its $\ell=2$ prototype (Barry, 2016, He, 5 Apr 2025, He, 2024).

1. Origins and conceptual setting

The immediate antecedent of the multiple almost-Riordan group is the almost-Riordan extension of the Riordan group. Barry, He, and Pantelidis defined the almost-Riordan group $a\mathcal R$ as a group of triples $(a,g,f)$, proved that it is a super group of the Riordan group, identified the normal subgroup $b$0, and showed that $b$1 (Barry, 2016). Closely related work treated higher-order almost-Riordan structures $b$2, $b$3, and beyond, where successive initial columns are made independent while a Riordan core remains in the interior (Barry et al., 2019).

A second precursor is the multiple Riordan group. He defined multiple Riordan type arrays and multiple Riordan arrays $b$4, together with a multiple Riordan semigroup and a multiple Riordan group, and characterized them by one $b$5-sequence and multiple $b$6-sequences (He, 5 Apr 2025). In that setting, a finite cycle of multiplier functions replaces the single Riordan multiplier $b$7, so that the column structure becomes periodic modulo $b$8.

The double almost-Riordan group supplied the direct model for the general theory. In the double case, the array $b$9 has columns $g$0; the paper established its group law, inverse, sequence characterization, production matrix, compression, and total-positivity results (He, 2024). The later multiple theory systematically replaces this 2-fold pattern by an $g$1-fold one (He, 2 Sep 2025).

2. Definition and column architecture

Fix $g$2. A multiple almost-Riordan array is built from

$g$3

and

$g$4

Its defining column generating functions are

$g$5

so the zeroth column is free, while every later column belongs to an $g$6-periodic multiple Riordan pattern (He, 2 Sep 2025).

Equivalently, the entries satisfy

$g$7

and for $g$8,

$g$9

This is the precise sense in which the construction is “almost”: the first column is governed by $\mathbb K[[t^\ell]]$0, whereas columns $\mathbb K[[t^\ell]]$1 are generated by the multiple Riordan body $\mathbb K[[t^\ell]]$2 (He, 2 Sep 2025).

The definition specializes correctly in lower ranks. When $\mathbb K[[t^\ell]]$3, it reduces to the classical almost-Riordan array $\mathbb K[[t^\ell]]$4 (He et al., 2024). When $\mathbb K[[t^\ell]]$5, it becomes the double almost-Riordan array

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

which was already studied as a group in its own right (He, 2024).

3. Fundamental theorem, group law, and internal structure

The effective substitution governing the theory is

$\mathbb K[[t^\ell]]$7

The first fundamental theorem for multiple almost-Riordan arrays acts residue class by residue class modulo $\mathbb K[[t^\ell]]$8. If

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

then

$f_j$0

If

$f_j$1

then

$f_j$2

More generally, for $f_j$3 and

$f_j$4

one has

$f_j$5

These formulas separate the exceptional first column from the periodic multiple Riordan action on the remaining residue classes (He, 2 Sep 2025).

From this operator calculus one obtains the group product. For

$f_j$6

the product is

$f_j$7

where $f_j$8 and $f_j$9 (He, 2 Sep 2025).

The identity element is

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

and the inverse of $t\mathbb K[[t^\ell]]$1 is

$t\mathbb K[[t^\ell]]$2

where $t\mathbb K[[t^\ell]]$3 and $t\mathbb K[[t^\ell]]$4 is the compositional inverse of $t\mathbb K[[t^\ell]]$5 (He, 2 Sep 2025).

The internal subgroup structure parallels the classical almost-Riordan picture. The multiple Appell subgroup

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

is normal, the multiple Lagrange subgroup is

$t\mathbb K[[t^\ell]]$7

and

$t\mathbb K[[t^\ell]]$8

The theory also includes Bell-type subgroups

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

together with related subgroups $(b,\; tg,\; tgf_1,\; tgf_1f_2,\dots)$0 defined by $(b,\; tg,\; tgf_1,\; tgf_1f_2,\dots)$1 (He, 2 Sep 2025).

4. Sequence characterization and production matrices

A defining feature of Riordan theory is its sequence characterization, and the multiple almost-Riordan group retains that principle in a higher-rank form. The array

$(b,\; tg,\; tgf_1,\; tgf_1f_2,\dots)$2

is decomposed into a free first-column part and $(b,\; tg,\; tgf_1,\; tgf_1f_2,\dots)$3 interlaced slices whose nonzero subarrays are ordinary Riordan arrays. After deleting zero columns and initial zero rows, these slices become

$(b,\; tg,\; tgf_1,\; tgf_1f_2,\dots)$4

up to the evident continuation through $(b,\; tg,\; tgf_1,\; tgf_1f_2,\dots)$5 (He, 2 Sep 2025). This decomposition is the mechanism behind the existence of one common $(b,\; tg,\; tgf_1,\; tgf_1f_2,\dots)$6-sequence, $(b,\; tg,\; tgf_1,\; tgf_1f_2,\dots)$7 $(b,\; tg,\; tgf_1,\; tgf_1f_2,\dots)$8-sequences, and an additional $(b,\; tg,\; tgf_1,\; tgf_1f_2,\dots)$9-sequence for the distinguished first column.

Let

$\ell=2$0

and let $\ell=2$1 with inverse $\ell=2$2. The basic structural identity is

$\ell=2$3

The paper also gives explicit formulas for $\ell=2$4 and $\ell=2$5 in terms of $\ell=2$6, $\ell=2$7, $\ell=2$8, $\ell=2$9, and the coefficients $a\mathcal R$0, with $a\mathcal R$1 (He, 2 Sep 2025). In particular, the multiple Riordan body contributes the common $a\mathcal R$2-data, while the “almost” deviation is carried by $a\mathcal R$3 and by the special form of $a\mathcal R$4.

The production matrix organizes these sequences in a single lower-Hessenberg object. Its column generating functions are

$a\mathcal R$5

This generalizes the production matrix of a classical Riordan array, where the columns are $a\mathcal R$6, and it simultaneously extends the $a\mathcal R$7-/$a\mathcal R$8-/$a\mathcal R$9-formalism developed for ordinary almost-Riordan arrays and the $(a,g,f)$0- plus multiple-$(a,g,f)$1-formalism of multiple Riordan arrays (He, 2 Sep 2025, He et al., 2024, He, 5 Apr 2025).

5. Compression and the double case as prototype

Multiple almost-Riordan arrays admit a compression operator that preserves the class. For

$(a,g,f)$2

its compression is

$(a,g,f)$3

The compressed array $(a,g,f)$4 is again a multiple almost-Riordan array

$(a,g,f)$5

where $(a,g,f)$6 are obtained by extracting the corresponding coefficient subsequences from $(a,g,f)$7 (He, 2 Sep 2025).

The compressed entries satisfy explicit formulas. The first column is

$(a,g,f)$8

and for $(a,g,f)$9 the remaining columns are described by products of $b$00 arranged in the same periodic pattern that defines the original group. The paper also provides a full sequence characterization of the compression, including functional equations for $b$01, $b$02, and $b$03 after replacing $b$04 by the compressed multiplier

$b$05

Thus compression is not merely a combinatorial thinning; it is an internal structural operation on the category of multiple almost-Riordan arrays (He, 2 Sep 2025).

The double almost-Riordan theory shows how this mechanism behaves in the first nontrivial case. For $b$06, the group $b$07 was defined, its first fundamental theorem was split according to parity, and its sequence characterization used one $b$08-sequence, two $b$09-sequences, and a $b$10-sequence; the production matrix had columns

$b$11

(He, 2024). The same paper defined compression by

$b$12

and showed that the compressed array satisfies recurrences expressed in the same sequence data.

Total positivity has been worked out in the ordinary and double almost-Riordan settings, but not yet in the general multiple almost-Riordan form in the supplied material. For ordinary almost-Riordan arrays, the production matrix $b$13 being totally positive implies total positivity of both $b$14 and $b$15, although the converse fails; tridiagonal $b$16 admits a necessary-and-sufficient criterion (He et al., 2024). For double almost-Riordan arrays, if $b$17 is totally positive, then $b$18 is totally positive for any $b$19, and $b$20 is totally positive when $b$21 (He, 2024). This suggests a natural multiple analogue through the $b$22-, $b$23-, and $b$24-data, but that step is not stated in the cited papers.

6. Neighboring Riordan-type groups and broader landscape

The multiple almost-Riordan group belongs to a broader family of multi-parameter Riordan-type groups, but its defining feature is specific: a distinguished first column attached to a multiple Riordan body. This distinguishes it from the Sprugnoli group, whose basic element is a triple $b$25 with column generating functions

$b$26

and from the triple Riordan group, whose elements are quadruples $b$27 governed by the effective substitution $b$28 and a 3-periodic column pattern (Barry, 15 May 2026, Barry, 2024). These groups are related in spirit, but they are not the same construction.

Other extensions move in different directions. Higher-dimensional Riordan groups replace one-variable series by $b$29-variable formal power series and identify the Riordan group with a holomorph $b$30, while bi-infinite Riordan matrices replace lower-triangular power-series matrices by Laurent-series matrices $b$31 indexed by $b$32 (O'Farrell, 2020, Prieto-Martínez et al., 10 Apr 2025). The Riordan near algebra furnishes a formal functional calculus and generalized powers for Riordan elements, showing that Riordan-type groups naturally sit inside larger algebraic environments (0902.2853).

Combinatorial motivation remains pervasive. The note “Notes on Riordan arrays and lattice paths” states that it explores links between Riordan arrays and lattice paths, includes almost Riordan arrays among the generalizations under discussion, and emphasizes the role of the $b$33-matrix characterization when downward steps are allowed (Barry, 13 Apr 2025). A plausible implication is that the multiple almost-Riordan group should support analogous lattice-path and production-matrix interpretations, especially because its characteristic data already organize the array into interlaced Riordan slices.

Within the current literature, the multiple almost-Riordan group is best understood as the point where three strands meet: the almost-Riordan enlargement of the Riordan group, the periodic column architecture of multiple Riordan arrays, and the sequence-theoretic machinery of $b$34-, $b$35-, and $b$36-data. Its formalization yields a noncommutative matrix group with explicit product, inverse, semidirect decomposition, sequence characterization, production matrix, and compression theory (He, 2 Sep 2025).