Generalized Riordan Arrays

• Generalized Riordan arrays are extensions of classical Riordan arrays, defined via formal series that act on generating functions through multiplication and substitution.
• They integrate diverse frameworks including weighted and q-dependent systems, higher-periodic constructions, and multivariate analogues to accommodate varied combinatorial and algebraic structures.
• These arrays underpin operator theory approaches and facilitate transformations such as production matrix analysis, linking them to orthogonal polynomials, lattice path models, and combinatorial matrix families.

Generalized Riordan arrays denote several extensions of the classical Riordan-array formalism in which lower-triangular matrices are encoded by formal series and act on generating functions by multiplication and substitution. In the classical setting, a Riordan array $(g(x),f(x))$ has entries $t_{n,k}=[x^n]\,g(x)f(x)^k$, group law $(g,f)(u,v)=\bigl(g(x)u(f(x)),\,v(f(x))\bigr)$, inverse $\left(\frac{1}{g(\bar f(x))},\bar f(x)\right)$, and bivariate generating function $\frac{g(x)}{1-yf(x)}$ (Barry, 2021). In current literature, the adjective “generalized” covers several non-equivalent enlargements of this template: weighted and $q$-dependent coefficient systems, generalized Pascal-matrix conjugations, higher-periodic groups using several substitution series, stretched and almost Riordan constructions, multivariate analogues, and semi-Laurent versions with bi-infinite matrices (Burlachenko, 2016, Zemel, 2015, Barry, 2024, Bugajewski et al., 4 Sep 2025). This suggests that the subject is best viewed as a family of compatible operator theories rather than a single universally fixed definition.

1. Classical core and operator-theoretic template

The common point of departure is the ordinary Riordan group. A Riordan array is specified by power series

$g(x)=g_0+g_1x+g_2x^2+\cdots,\qquad g_0\neq 0,$

and

$f(x)=f_1x+f_2x^2+\cdots,\qquad f_1\neq 0,$

with entries $t_{n,k}=[x^n]\,g(x)f(x)^k$. Its action on a series $h(x)$ is the Fundamental Theorem of Riordan arrays,

$t_{n,k}=[x^n]\,g(x)f(x)^k$0

and the associated production matrix is lower Hessenberg and determined by the $t_{n,k}=[x^n]\,g(x)f(x)^k$1- and $t_{n,k}=[x^n]\,g(x)f(x)^k$2-sequences. In the standard characterization recalled in the production-matrix literature, if $t_{n,k}=[x^n]\,g(x)f(x)^k$3 and $t_{n,k}=[x^n]\,g(x)f(x)^k$4, then

$t_{n,k}=[x^n]\,g(x)f(x)^k$5

This classical skeleton remains the reference point for most generalized constructions (Barry, 2020).

A second foundational thread is the operator and umbral viewpoint. In the symbolic treatment of Riordan arrays, exponential Riordan arrays are represented by umbral pairs $t_{n,k}=[x^n]\,g(x)f(x)^k$6, and the generalized $t_{n,k}=[x^n]\,g(x)f(x)^k$7-Riordan array attached to a sequence $t_{n,k}=[x^n]\,g(x)f(x)^k$8 is given by

$t_{n,k}=[x^n]\,g(x)f(x)^k$9

This recovers exponential Riordan arrays when $(g,f)(u,v)=\bigl(g(x)u(f(x)),\,v(f(x))\bigr)$0 and ordinary Riordan arrays when $(g,f)(u,v)=\bigl(g(x)u(f(x)),\,v(f(x))\bigr)$1. The same framework derives the group law, inverse, Lagrange inversion, nonrecursive entry formulas, and generalized recurrences from Abel-type identities, rather than taking generating-function manipulations as primary (Agapito et al., 2015).

The persistence of the same two operations—multiplication by an invertible prefactor and substitution by a compositional unit—is the organizing principle across the later generalizations. Even when the ambient algebra, indexing set, or coefficient basis changes, the generalized object is typically designed so that matrix multiplication still represents an operator of “multiply, then substitute” type.

2. Weighted, $(g,f)(u,v)=\bigl(g(x)u(f(x)),\,v(f(x))\bigr)$2-dependent, and generalized Pascal-matrix frameworks

One major route to generalization changes the coefficient algebra underlying the array. A generalized binomial system is built from a sequence $(g,f)(u,v)=\bigl(g(x)u(f(x)),\,v(f(x))\bigr)$3 with $(g,f)(u,v)=\bigl(g(x)u(f(x)),\,v(f(x))\bigr)$4, generalized factorials

$(g,f)(u,v)=\bigl(g(x)u(f(x)),\,v(f(x))\bigr)$5

and generalized binomial coefficients

$(g,f)(u,v)=\bigl(g(x)u(f(x)),\,v(f(x))\bigr)$6

The associated generalized Pascal matrix $(g,f)(u,v)=\bigl(g(x)u(f(x)),\,v(f(x))\bigr)$7 is central, and generalized Riordan arrays are obtained by diagonal conjugation,

$(g,f)(u,v)=\bigl(g(x)u(f(x)),\,v(f(x))\bigr)$8

In this setting, the parameter $(g,f)(u,v)=\bigl(g(x)u(f(x)),\,v(f(x))\bigr)$9 organizes a distinguished family: $\left(\frac{1}{g(\bar f(x))},\bar f(x)\right)$0 gives ordinary Riordan arrays, $\left(\frac{1}{g(\bar f(x))},\bar f(x)\right)$1 gives exponential Riordan arrays, $\left(\frac{1}{g(\bar f(x))},\bar f(x)\right)$2 gives arrays associated with $\left(\frac{1}{g(\bar f(x))},\bar f(x)\right)$3-binomial coefficients, and $\left(\frac{1}{g(\bar f(x))},\bar f(x)\right)$4 is exceptional and leads to a separate zero-generalized-Pascal theory (Burlachenko, 2016).

A parallel formulation replaces the coefficient sequence $\left(\frac{1}{g(\bar f(x))},\bar f(x)\right)$5 by a general weight

$\left(\frac{1}{g(\bar f(x))},\bar f(x)\right)$6

and defines $\left(\frac{1}{g(\bar f(x))},\bar f(x)\right)$7-Riordan arrays through weighted column series. In the standard form, a $\left(\frac{1}{g(\bar f(x))},\bar f(x)\right)$8-Riordan array is represented by

$\left(\frac{1}{g(\bar f(x))},\bar f(x)\right)$9

and the resulting group $\frac{g(x)}{1-yf(x)}$0 is isomorphic to

$\frac{g(x)}{1-yf(x)}$1

The operator interpretation is equally important: $\frac{g(x)}{1-yf(x)}$2-Riordan arrays are exactly the matrices of $\frac{g(x)}{1-yf(x)}$3-Sheffer sequences, and all groups arising from different weights are conjugate inside the full lower-triangular group of graded polynomial sequences (Zemel, 2015).

The zero-generalized-Pascal extensions enlarge the subject further by admitting limiting or degenerate central matrices such as $\frac{g(x)}{1-yf(x)}$4 and the fractal products

$\frac{g(x)}{1-yf(x)}$5

These constructions include the $\frac{g(x)}{1-yf(x)}$6 case and Pascal triangle modulo $\frac{g(x)}{1-yf(x)}$7, and the associated matrix groups preserve a Riordan-like multiplication law with a modified algebra product. An analogue of the Lagrange inversion theorem is developed in this setting, and zero generalized Pascal matrices are interpreted as “points at infinity” relative to the space of generalized Pascal matrices (Burlachenko, 2021).

Framework	Defining data	Distinguished specializations
$\frac{g(x)}{1-yf(x)}$8-Riordan arrays	Sequence $\frac{g(x)}{1-yf(x)}$9, conjugation by $q$0	ordinary, exponential, $q$1-binomial
$q$2-Riordan arrays	Weight $q$3	weighted Sheffer/Riordan theories
Zero generalized Pascal setting	$q$4, $q$5	$q$6, Pascal triangle modulo $q$7

These formulations are structurally close but not identical. A plausible implication is that “generalized Riordan array” functions as an umbrella term for extensions that preserve some Riordan mechanisms while varying the underlying coefficient geometry.

3. Higher-periodic, bisection-based, and multivariate generalizations

Another route to generalization replaces the single substitution series $q$8 by several series interleaved in a periodic pattern. The triple Riordan group $q$9 consists of $g(x)=g_0+g_1x+g_2x^2+\cdots,\qquad g_0\neq 0,$0-tuples

$g(x)=g_0+g_1x+g_2x^2+\cdots,\qquad g_0\neq 0,$1

with

$g(x)=g_0+g_1x+g_2x^2+\cdots,\qquad g_0\neq 0,$2

and product law

$g(x)=g_0+g_1x+g_2x^2+\cdots,\qquad g_0\neq 0,$3

where

$g(x)=g_0+g_1x+g_2x^2+\cdots,\qquad g_0\neq 0,$4

Its bivariate generating function is

$g(x)=g_0+g_1x+g_2x^2+\cdots,\qquad g_0\neq 0,$5

and the specialization $g(x)=g_0+g_1x+g_2x^2+\cdots,\qquad g_0\neq 0,$6 embeds an ordinary Riordan array into the triple theory. The paper presents this as the cubic continuation of the double Riordan construction and as a template for further $g(x)=g_0+g_1x+g_2x^2+\cdots,\qquad g_0\neq 0,$7-fold periodic groups (Barry, 2024).

A different higher-order extension is the Sprugnoli group, whose elements are triples $g(x)=g_0+g_1x+g_2x^2+\cdots,\qquad g_0\neq 0,$8 with $g(x)=g_0+g_1x+g_2x^2+\cdots,\qquad g_0\neq 0,$9, $f(x)=f_1x+f_2x^2+\cdots,\qquad f_1\neq 0,$0, and $f(x)=f_1x+f_2x^2+\cdots,\qquad f_1\neq 0,$1. Here the columns alternate according to even and odd bisections, and the fundamental action on a series $f(x)=f_1x+f_2x^2+\cdots,\qquad f_1\neq 0,$2 is

$f(x)=f_1x+f_2x^2+\cdots,\qquad f_1\neq 0,$3

where

$f(x)=f_1x+f_2x^2+\cdots,\qquad f_1\neq 0,$4

The bivariate generating function is

$f(x)=f_1x+f_2x^2+\cdots,\qquad f_1\neq 0,$5

the identity is $f(x)=f_1x+f_2x^2+\cdots,\qquad f_1\neq 0,$6, and the production matrix involves alternating $f(x)=f_1x+f_2x^2+\cdots,\qquad f_1\neq 0,$7- and $f(x)=f_1x+f_2x^2+\cdots,\qquad f_1\neq 0,$8-stripe data rather than a single $f(x)=f_1x+f_2x^2+\cdots,\qquad f_1\neq 0,$9-sequence (Barry, 15 May 2026).

A genuinely different notion of higher generalization is multivariate. In $t_{n,k}=[x^n]\,g(x)f(x)^k$0 variables, one works in

$t_{n,k}=[x^n]\,g(x)f(x)^k$1

with units $t_{n,k}=[x^n]\,g(x)f(x)^k$2 and the group $t_{n,k}=[x^n]\,g(x)f(x)^k$3 of formal maps $t_{n,k}=[x^n]\,g(x)f(x)^k$4 fixing the origin and compositional invertible. The higher-dimensional Riordan group is

$t_{n,k}=[x^n]\,g(x)f(x)^k$5

with law

$t_{n,k}=[x^n]\,g(x)f(x)^k$6

and matrices are indexed by monic monomials rather than by integers. The corresponding multivariate Fundamental Theorem states that matrix multiplication exactly matches multiplication in $t_{n,k}=[x^n]\,g(x)f(x)^k$7 together with composition by formal automorphisms (O'Farrell, 2020).

These constructions represent different meanings of “higher-order.” The triple and Sprugnoli groups keep one formal variable but impose periodic or parity structure; the multivariate groups replace the one-variable indexing entirely.

4. Structural transformations, characterizations, and near-Riordan objects

Generalized Riordan theory is also driven by transformations that preserve, deform, or reconstruct Riordan structure. The production-matrix approach is central here. For an ordinary Riordan array $t_{n,k}=[x^n]\,g(x)f(x)^k$8, the ordinary production matrix is lower Hessenberg and determined by the $t_{n,k}=[x^n]\,g(x)f(x)^k$9- and $h(x)$0-data; the second production matrix is obtained by deleting the first column of the ordinary production matrix, and the matrix generated by it is again a Riordan array with components expressed explicitly in terms of $h(x)$1, $h(x)$2, $h(x)$3, and $h(x)$4. The same pattern is extended to third production matrices and conjecturally to all $h(x)$5-th production matrices, yielding a canonical hierarchy attached to a single initial Riordan array (Barry, 2020).

Shifted central extraction gives another systematic construction. If

$h(x)$6

and

$h(x)$7

then every $h(x)$8-shifted central triangle is itself a Riordan array, with factorization

$h(x)$9

The family satisfies transition relations independent of $t_{n,k}=[x^n]\,g(x)f(x)^k$00, and the generating function of the $t_{n,k}=[x^n]\,g(x)f(x)^k$01-sequence of $t_{n,k}=[x^n]\,g(x)f(x)^k$02 is $t_{n,k}=[x^n]\,g(x)f(x)^k$03 when $t_{n,k}=[x^n]\,g(x)f(x)^k$04 is the $t_{n,k}=[x^n]\,g(x)f(x)^k$05-sequence generating function of the original array (Barry, 2019).

Partial summation shows how Riordan structure can persist or fail under natural matrix operations. Column partial sums of $t_{n,k}=[x^n]\,g(x)f(x)^k$06 produce the Riordan array

$t_{n,k}=[x^n]\,g(x)f(x)^k$07

whereas row partial sums generally leave the Riordan group and yield a rectangular matrix whose inverse is lower Hessenberg. A further variant leads to an almost Riordan array linked in the paper to Fibonacci numbers and phyllotaxis, showing that generalized Riordan behavior can survive outside the strict lower-triangular group (Barry, 2021).

The $t_{n,k}=[x^n]\,g(x)f(x)^k$08-matrix viewpoint sharpens these structural statements. Riordan arrays can be characterized by local recurrences encoded by an $t_{n,k}=[x^n]\,g(x)f(x)^k$09-matrix and a $t_{n,k}=[x^n]\,g(x)f(x)^k$10-sequence, and sparse $t_{n,k}=[x^n]\,g(x)f(x)^k$11-matrices are used to construct Bell-type matrices, quasi-involutions, and moment matrices of perturbed orthogonal polynomials. In particular, certain two-row $t_{n,k}=[x^n]\,g(x)f(x)^k$12-matrices are conjectured to produce first-column Hankel transforms that satisfy Somos $t_{n,k}=[x^n]\,g(x)f(x)^k$13 recurrences (Barry, 2019).

Recent work on lattice paths indicates another structural synthesis. A 2025 note states that it describes Riordan arrays and some of their generalizations, including rectifications and triangulations, links Riordan arrays to lattice paths with steps of type $t_{n,k}=[x^n]\,g(x)f(x)^k$14, treats almost Riordan arrays in lattice-path models, and uses the $t_{n,k}=[x^n]\,g(x)f(x)^k$15-matrix characterization to analyze path families with downward steps (Barry, 13 Apr 2025).

5. Polynomial triangles, Eulerian transforms, and combinatorial matrix families

Generalized Riordan arrays frequently appear as coefficient arrays of polynomial systems and as transformation devices between classical combinatorial triangles. One important example is the theory of generalized Euler polynomials. For a polynomial sequence $t_{n,k}=[x^n]\,g(x)f(x)^k$16 of degree $t_{n,k}=[x^n]\,g(x)f(x)^k$17,

$t_{n,k}=[x^n]\,g(x)f(x)^k$18

is realized as the normalized row polynomial of a square or generalized Riordan array $t_{n,k}=[x^n]\,g(x)f(x)^k$19. The matrices $t_{n,k}=[x^n]\,g(x)f(x)^k$20 and $t_{n,k}=[x^n]\,g(x)f(x)^k$21 mediate transformations involving $t_{n,k}=[x^n]\,g(x)f(x)^k$22, $t_{n,k}=[x^n]\,g(x)f(x)^k$23, Stirling numbers, multinomial coefficients, Bell polynomials, and generalized Lagrange series. In this formulation, classical Euler polynomials arise as the special case $t_{n,k}=[x^n]\,g(x)f(x)^k$24 (Burlachenko, 2017).

Pascal-like triangles form another major branch. For the ordinary Riordan family

$t_{n,k}=[x^n]\,g(x)f(x)^k$25

the rows are palindromic and admit $t_{n,k}=[x^n]\,g(x)f(x)^k$26-vector decompositions. The associated $t_{n,k}=[x^n]\,g(x)f(x)^k$27-matrix is the stretched Riordan array

$t_{n,k}=[x^n]\,g(x)f(x)^k$28

with entries

$t_{n,k}=[x^n]\,g(x)f(x)^k$29

Reversion of these Pascal-like triangles produces generalized Narayana families, and their generating functions are expressed by Jacobi continued fractions (Barry, 2018).

The related $t_{n,k}=[x^n]\,g(x)f(x)^k$30-matrix theory studies the binomial multiplication

$t_{n,k}=[x^n]\,g(x)f(x)^k$31

where $t_{n,k}=[x^n]\,g(x)f(x)^k$32 is the Pascal matrix. For the same ordinary Riordan family, the $t_{n,k}=[x^n]\,g(x)f(x)^k$33-matrix is

$t_{n,k}=[x^n]\,g(x)f(x)^k$34

while in the exponential case

$t_{n,k}=[x^n]\,g(x)f(x)^k$35

The paper emphasizes that the $t_{n,k}=[x^n]\,g(x)f(x)^k$36-, $t_{n,k}=[x^n]\,g(x)f(x)^k$37-, and $t_{n,k}=[x^n]\,g(x)f(x)^k$38-matrices are linked by systematic transformations of the coefficients in corresponding Jacobi continued fractions (Barry, 2018).

A further parametric deformation is obtained by multiplying any Riordan array $t_{n,k}=[x^n]\,g(x)f(x)^k$39 on the right by the generalized Pascal array

$t_{n,k}=[x^n]\,g(x)f(x)^k$40

producing

$t_{n,k}=[x^n]\,g(x)f(x)^k$41

For the Catalan array

$t_{n,k}=[x^n]\,g(x)f(x)^k$42

this yields the $t_{n,k}=[x^n]\,g(x)f(x)^k$43-Catalan triangle

$t_{n,k}=[x^n]\,g(x)f(x)^k$44

which interpolates between the Aigner array at $t_{n,k}=[x^n]\,g(x)f(x)^k$45 and the Shapiro array at $t_{n,k}=[x^n]\,g(x)f(x)^k$46. The associated Sheffer polynomials satisfy the shift relation

$t_{n,k}=[x^n]\,g(x)f(x)^k$47

and specializations lead to identities involving Chebyshev polynomials of the second kind, periodic sequences, and Fibonacci numbers (Agapito et al., 2015).

6. Orthogonal-polynomial, Hankel-transform, and semi-Laurent directions

A substantial part of generalized Riordan theory lies at the interface with orthogonal and biorthogonal polynomials. Laurent biorthogonal polynomials with constant-coefficient three-term recurrence

$t_{n,k}=[x^n]\,g(x)f(x)^k$48

have coefficient arrays that are Riordan arrays, for example

$t_{n,k}=[x^n]\,g(x)f(x)^k$49

or, in the preferred normalization,

$t_{n,k}=[x^n]\,g(x)f(x)^k$50

The inverse Riordan array gives a natural associated family of ordinary orthogonal polynomials, and the moment Hankel transform is

$t_{n,k}=[x^n]\,g(x)f(x)^k$51

The theory extends to a three-parameter generalized recurrence

$t_{n,k}=[x^n]\,g(x)f(x)^k$52

which still admits a Riordan-array description (Barry, 2013).

Catalan-based recurrence solving is another prominent application. Generalized Catalan-Schroeder convolution recurrences are solved by writing their generating functions as Riordan transforms of the Catalan series

$t_{n,k}=[x^n]\,g(x)f(x)^k$53

The resulting sequences are often moment sequences for orthogonal polynomials, and many of their Hankel transforms are conjectured or shown to satisfy Somos $t_{n,k}=[x^n]\,g(x)f(x)^k$54 recurrences. The same framework is used to connect Riordan arrays with elliptic curves, continued fractions, and orthogonal-polynomial moment matrices (Barry, 2019).

Stretched Riordan arrays amplify this Hankel-transform direction. In conjectural families built from generalized Jacobi continued fractions and Catalan numbers, coefficient sequences are described by stretched Riordan arrays of the form $t_{n,k}=[x^n]\,g(x)f(x)^k$55, and their Hankel transforms are conjectured to be Somos $t_{n,k}=[x^n]\,g(x)f(x)^k$56, Somos $t_{n,k}=[x^n]\,g(x)f(x)^k$57, or Somos $t_{n,k}=[x^n]\,g(x)f(x)^k$58 sequences. In this setting, the Riordan array provides the coefficient-extraction mechanism, Catalan numbers provide the seed sequence, and the Hankel transform supplies the bridge to bilinear integrable recurrences (Barry, 2022).

A more radical generalization replaces formal power series by formal semi-Laurent series. If $t_{n,k}=[x^n]\,g(x)f(x)^k$59 is a formal semi-Laurent series and $t_{n,k}=[x^n]\,g(x)f(x)^k$60 is a formal power series with $t_{n,k}=[x^n]\,g(x)f(x)^k$61, the generalized Riordan array, or Riordan Laurent matrix, is the bi-infinite matrix

$t_{n,k}=[x^n]\,g(x)f(x)^k$62

The multiplication law is

$t_{n,k}=[x^n]\,g(x)f(x)^k$63

with inverse

$t_{n,k}=[x^n]\,g(x)f(x)^k$64

The order-zero subgroup $t_{n,k}=[x^n]\,g(x)f(x)^k$65 is identified with the classical Riordan group, but the generalized group is proved not to be isomorphic to the classical one. Over $t_{n,k}=[x^n]\,g(x)f(x)^k$66, the same paper constructs an infinite-dimensional Lie group modeled on $t_{n,k}=[x^n]\,g(x)f(x)^k$67 and a Lie algebra

$t_{n,k}=[x^n]\,g(x)f(x)^k$68

with bracket

$t_{n,k}=[x^n]\,g(x)f(x)^k$69

This places generalized Riordan arrays in a setting that is simultaneously combinatorial, algebraic, and Lie-theoretic (Bugajewski et al., 4 Sep 2025).

One persistent misconception is that all generalized Riordan theories are equivalent reformulations of the same object. The literature does not support that view. Weighted $t_{n,k}=[x^n]\,g(x)f(x)^k$70-Riordan groups are conjugate for different weights (Zemel, 2015), but semi-Laurent generalized Riordan groups are proved not to be isomorphic to the classical Riordan group (Bugajewski et al., 4 Sep 2025), and periodic constructions such as the triple and Sprugnoli groups introduce genuinely different column-generation mechanisms (Barry, 2024, Barry, 15 May 2026). The modern subject is therefore unified by method and operator form, not by a single definition.