Riordan Matrices: Theory and Applications

• Riordan matrices are infinite lower triangular matrices defined by formal power series that represent weighted substitutions and generating-function transforms.
• They form a structured group via matrix multiplication, reflecting key properties of series composition, inverse elements, and finite-order behaviour.
• Applications span combinatorial enumeration, algebraic structures, and graph-theoretic models, linking Pascal-like triangles to weighted matrices.

Riordan matrices are infinite lower triangular matrices determined by formal power series and interpreted simultaneously as matrix objects, operator-theoretic substitutions, and generating-function transforms. In the ordinary setting, a Riordan array $(g,f)$ has entries $d_{n,k}=[x^n]\,g(x)f(x)^k$; in the exponential setting, $[g,f]$ has entries $d_{n,k}=\frac{n!}{k!}[x^n]\,g(x)f(x)^k$. Across these forms, Riordan theory organizes multiplication and composition of series, Sheffer-type polynomial systems, Pascal-like triangles, and a wide range of generalizations to weighted, multivariate, bi-infinite, and parity-sensitive matrix groups (Frankson, 2024, Barry, 2018, O'Farrell, 2020).

1. Classical definitions and equivalent descriptions

An ordinary Riordan array is an invertible lower-triangular matrix determined by formal power series

$$g(x)=1+g_1x+g_2x^2+\cdots,\qquad f(x)=x+f_2x^2+f_3x^3+\cdots,$$

with entries

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

The $k$-th column generating function is $g(x)f(x)^k$, the bivariate generating function is

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

the product law is

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

and the inverse is

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

where $d_{n,k}=[x^n]\,g(x)f(x)^k$1 is the compositional inverse of $d_{n,k}=[x^n]\,g(x)f(x)^k$2. The Fundamental Theorem of Riordan Arrays identifies matrix multiplication with weighted substitution: $d_{n,k}=[x^n]\,g(x)f(x)^k$3 This formalism is the standard definition of the Riordan group $d_{n,k}=[x^n]\,g(x)f(x)^k$4 (Frankson, 2024).

The exponential analogue replaces ordinary column extraction by

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

with bivariate exponential generating function

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

The binomial matrix appears in both settings: $d_{n,k}=[x^n]\,g(x)f(x)^k$7 This dual appearance is central in applications to $d_{n,k}=[x^n]\,g(x)f(x)^k$8-, $d_{n,k}=[x^n]\,g(x)f(x)^k$9-, and $[g,f]$0-matrices of Pascal-like triangles (Barry, 2018).

A further equivalent notation, used heavily in diagonal and palindromicity theory, is

$[g,f]$1

with entry formula

$[g,f]$2

In this notation the operator action is

$[g,f]$3

which is again a weighted substitution operator (Cheon et al., 16 Feb 2026).

2. Group-theoretic framework

Over a field of characteristic $[g,f]$4, the Riordan group may be written as the semidirect product

$[g,f]$5

with multiplication

$[g,f]$6

The $[g,f]$7-th power is

$[g,f]$8

so finite order is governed simultaneously by multiplicative order in the first coordinate and compositional order in the second (Cohen, 2018).

One extension is the Double Riordan group $[g,f]$9, whose elements are triples $d_{n,k}=\frac{n!}{k!}[x^n]\,g(x)f(x)^k$0 with $d_{n,k}=\frac{n!}{k!}[x^n]\,g(x)f(x)^k$1 even and $d_{n,k}=\frac{n!}{k!}[x^n]\,g(x)f(x)^k$2 odd. Its columns alternate according to

$d_{n,k}=\frac{n!}{k!}[x^n]\,g(x)f(x)^k$3

and its group law is

$d_{n,k}=\frac{n!}{k!}[x^n]\,g(x)f(x)^k$4

A central 2024 result identifies an isomorphic copy of the full Riordan group inside $d_{n,k}=\frac{n!}{k!}[x^n]\,g(x)f(x)^k$5: the map

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

is a group monomorphism onto the type-1 almost Appell subgroup (Frankson, 2024).

The same semidirect-product pattern persists in several variables. For $d_{n,k}=\frac{n!}{k!}[x^n]\,g(x)f(x)^k$7 commuting variables, the higher-dimensional Riordan group is

$d_{n,k}=\frac{n!}{k!}[x^n]\,g(x)f(x)^k$8

where $d_{n,k}=\frac{n!}{k!}[x^n]\,g(x)f(x)^k$9 and $$g(x)=1+g_1x+g_2x^2+\cdots,\qquad f(x)=x+f_2x^2+f_3x^3+\cdots,$$0 is the group of compositionally invertible $$g(x)=1+g_1x+g_2x^2+\cdots,\qquad f(x)=x+f_2x^2+f_3x^3+\cdots,$$1-tuples. Its matrix entries are indexed by monomials $$g(x)=1+g_1x+g_2x^2+\cdots,\qquad f(x)=x+f_2x^2+f_3x^3+\cdots,$$2 and satisfy

$$g(x)=1+g_1x+g_2x^2+\cdots,\qquad f(x)=x+f_2x^2+f_3x^3+\cdots,$$3

so the classical one-variable theory reappears as the $$g(x)=1+g_1x+g_2x^2+\cdots,\qquad f(x)=x+f_2x^2+f_3x^3+\cdots,$$4 case (O'Farrell, 2020).

A different generalization replaces the classical kernel by an arbitrary weight

$$g(x)=1+g_1x+g_2x^2+\cdots,\qquad f(x)=x+f_2x^2+f_3x^3+\cdots,$$5

The resulting weighted Riordan groups $$g(x)=1+g_1x+g_2x^2+\cdots,\qquad f(x)=x+f_2x^2+f_3x^3+\cdots,$$6 consist of lower triangular matrices whose associated bivariate series has the form

$$g(x)=1+g_1x+g_2x^2+\cdots,\qquad f(x)=x+f_2x^2+f_3x^3+\cdots,$$7

All such groups are conjugate inside one ambient lower-triangular group, so the ordinary and exponential Riordan groups are different realizations of one generalized structure (Zemel, 2015).

3. Structural descriptions beyond the column definition

Classical Riordan theory is often organized by the $$g(x)=1+g_1x+g_2x^2+\cdots,\qquad f(x)=x+f_2x^2+f_3x^3+\cdots,$$8-sequence. If $$g(x)=1+g_1x+g_2x^2+\cdots,\qquad f(x)=x+f_2x^2+f_3x^3+\cdots,$$9 and $a_{n,k}=[x^n]\,g(x)f(x)^k.$0 satisfies

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

then the entries obey the horizontal recursion

$a_{n,k}=[x^n]\,g(x)f(x)^k.$2

This is the standard row-wise propagation mechanism (Cheon et al., 16 Feb 2026).

A recent complementary viewpoint is diagonal. For the $a_{n,k}=[x^n]\,g(x)f(x)^k.$3-th diagonal

$a_{n,k}=[x^n]\,g(x)f(x)^k.$4

the diagonal recurrence is

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

and the associated bivariate generating function is

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

This gives a full diagonal-wise description for every ordinary Riordan matrix $a_{n,k}=[x^n]\,g(x)f(x)^k.$7, and it leads to a complete characterization of palindromic Riordan matrices (Cheon et al., 16 Feb 2026).

A second diagonal theory studies numerator polynomials. For ordinary Riordan matrices, the generating function of a fixed diagonal has numerator $a_{n,k}=[x^n]\,g(x)f(x)^k.$8, the generalized Euler polynomial; for exponential Riordan matrices, the corresponding numerator is $a_{n,k}=[x^n]\,g(x)f(x)^k.$9, the generalized Narayana polynomial. The paper introducing these families constructs explicit finite-dimensional transforms

$k$0

so that $k$1 maps the ordinary diagonal numerators $k$2 to the exponential diagonal numerators $k$3. This turns diagonal extraction into a finite operator calculus tied to Stirling-number bases and generalized Lagrange deformations $k$4 (Burlachenko, 2020).

A further structural complement is the vertical recursive relation. For a Riordan array $k$5,

$k$6

This column-wise recursion is encoded by quasi-Riordan matrices

$k$7

which satisfy

$k$8

The quasi-Riordan arrays form a group with multiplication

$k$9

providing a matrix formalism for vertical propagation and for finite-order truncation transforms (He, 2022).

4. Finite order, diagonalization, and eigenvectors

Finite-order elements of the Riordan group are rigid. If $g(x)f(x)^k$0 has finite compositional order $g(x)f(x)^k$1 and $g(x)f(x)^k$2 has finite multiplicative order $g(x)f(x)^k$3, then a Riordan element $g(x)f(x)^k$4 has finite order

$g(x)f(x)^k$5

precisely when

$g(x)f(x)^k$6

for some $g(x)f(x)^k$7. Every finite-order Riordan element is conjugate in $g(x)f(x)^k$8 to the diagonal form

$g(x)f(x)^k$9

where $\frac{g(x)}{1-yf(x)},$0 is a root of unity, and the conjugacy class is completely determined by $\frac{g(x)}{1-yf(x)},$1 (Cohen, 2018).

The same structure yields an explicit eigenvector theory. If $\frac{g(x)}{1-yf(x)},$2, $\frac{g(x)}{1-yf(x)},$3, and $\frac{g(x)}{1-yf(x)},$4 is the standard conjugating series with

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

then every eigenvector generating function has the form

$\frac{g(x)}{1-yf(x)},$6

where

$\frac{g(x)}{1-yf(x)},$7

and the corresponding eigenvalue is

$\frac{g(x)}{1-yf(x)},$8

Thus finite-order Riordan matrices are similar to periodic diagonal matrices, and their eigenspaces are parameterized by residue classes modulo $\frac{g(x)}{1-yf(x)},$9 (Cohen, 2018).

For general Riordan matrices, diagonalization can be characterized intrinsically. If $(g,f)\cdot(u,v)=\bigl(g\,u(f),\,v(f)\bigr),$0 and $(g,f)\cdot(u,v)=\bigl(g\,u(f),\,v(f)\bigr),$1, then

$(g,f)\cdot(u,v)=\bigl(g\,u(f),\,v(f)\bigr),$2

if and only if

$(g,f)\cdot(u,v)=\bigl(g\,u(f),\,v(f)\bigr),$3

A decisive consequence is that a Riordan matrix is diagonalizable in the full lower-triangular group $(g,f)\cdot(u,v)=\bigl(g\,u(f),\,v(f)\bigr),$4 if and only if it is diagonalizable inside the Riordan group itself (Cheon et al., 2021).

Pseudo-involutions supply a separate linear-algebraic phenomenon. For an $(g,f)\cdot(u,v)=\bigl(g\,u(f),\,v(f)\bigr),$5 truncation $(g,f)\cdot(u,v)=\bigl(g\,u(f),\,v(f)\bigr),$6, if $(g,f)\cdot(u,v)=\bigl(g\,u(f),\,v(f)\bigr),$7 is an involution with $(g,f)\cdot(u,v)=\bigl(g\,u(f),\,v(f)\bigr),$8, then the singular values of $(g,f)\cdot(u,v)=\bigl(g\,u(f),\,v(f)\bigr),$9 occur in reciprocal pairs. The proof uses the similarity of $d_{n,k}=[x^n]\,g(x)f(x)^k$00 and $d_{n,k}=[x^n]\,g(x)f(x)^k$01, so the spectrum of the positive matrix $d_{n,k}=[x^n]\,g(x)f(x)^k$02 is invariant under inversion (Cheon et al., 2021).

5. Combinatorial, geometric, and graph-theoretic applications

A major application is to Pascal-like triangles and their $d_{n,k}=[x^n]\,g(x)f(x)^k$03-matrices. If $d_{n,k}=[x^n]\,g(x)f(x)^k$04 is a Pascal-like matrix, its $d_{n,k}=[x^n]\,g(x)f(x)^k$05-matrix is defined uniformly by

$d_{n,k}=[x^n]\,g(x)f(x)^k$06

where $d_{n,k}=[x^n]\,g(x)f(x)^k$07 is the binomial matrix. For the ordinary family

$d_{n,k}=[x^n]\,g(x)f(x)^k$08

the associated $d_{n,k}=[x^n]\,g(x)f(x)^k$09-matrix is

$d_{n,k}=[x^n]\,g(x)f(x)^k$10

with reversed bivariate generating function

$d_{n,k}=[x^n]\,g(x)f(x)^k$11

For the exponential family

$d_{n,k}=[x^n]\,g(x)f(x)^k$12

the $d_{n,k}=[x^n]\,g(x)f(x)^k$13-matrix is

$d_{n,k}=[x^n]\,g(x)f(x)^k$14

In the exponential case, the reversed generating functions of the $d_{n,k}=[x^n]\,g(x)f(x)^k$15-, $d_{n,k}=[x^n]\,g(x)f(x)^k$16-, and $d_{n,k}=[x^n]\,g(x)f(x)^k$17-matrices admit explicit Jacobi continued fractions, giving a compact analytic description of the $d_{n,k}=[x^n]\,g(x)f(x)^k$18 transformation (Barry, 2018).

Palindromicity is especially transparent in the diagonal formalism. A Riordan matrix $d_{n,k}=[x^n]\,g(x)f(x)^k$19 is palindromic exactly when

$d_{n,k}=[x^n]\,g(x)f(x)^k$20

equivalently,

$d_{n,k}=[x^n]\,g(x)f(x)^k$21

The complete classification is

$d_{n,k}=[x^n]\,g(x)f(x)^k$22

and the resulting row polynomials admit a second-order recurrence. The same paper gives a quotient-of-Delannoy-paths model realizing this entire palindromic family combinatorially (Cheon et al., 16 Feb 2026).

Another specialized family is

$d_{n,k}=[x^n]\,g(x)f(x)^k$23

with $d_{n,k}=[x^n]\,g(x)f(x)^k$24 a degree-$d_{n,k}=[x^n]\,g(x)f(x)^k$25 polynomial. Each column is eventually periodic with period $d_{n,k}=[x^n]\,g(x)f(x)^k$26, and periodicity begins at

$d_{n,k}=[x^n]\,g(x)f(x)^k$27

in the $d_{n,k}=[x^n]\,g(x)f(x)^k$28-th column. The repeating tail blocks evolve under the powers of a circulant matrix

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

so the long-term column behavior is a finite-dimensional linear dynamical system (Krylov, 2023).

Riordan matrices also encode oriented graphs. An oriented Riordan graph $d_{n,k}=[x^n]\,g(x)f(x)^k$30 has skew-adjacency matrix

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

Modulo $d_{n,k}=[x^n]\,g(x)f(x)^k$32, the lower-triangular Riordan data determine the full orientation pattern. The induced subgraphs on residue classes modulo $d_{n,k}=[x^n]\,g(x)f(x)^k$33 are again oriented Riordan graphs, and the $d_{n,k}=[x^n]\,g(x)f(x)^k$34-sequence drives a fractal property: translated vertex intervals can induce isomorphic oriented subgraphs at powers of $d_{n,k}=[x^n]\,g(x)f(x)^k$35 (Jung, 2020).

6. Broader generalizations and recent directions

Several recent extensions enlarge the ambient algebra. One replaces formal power series by formal Laurent series and lower-triangular arrays by bi-infinite matrices indexed by $d_{n,k}=[x^n]\,g(x)f(x)^k$36. In that setting, a bi-infinite Riordan matrix $d_{n,k}=[x^n]\,g(x)f(x)^k$37 is defined by the columns of

$d_{n,k}=[x^n]\,g(x)f(x)^k$38

and acts by

$d_{n,k}=[x^n]\,g(x)f(x)^k$39

whenever the Laurent-series composition is well defined. This yields generalized Toeplitz and Lagrange subgroups, admits invertible substitutions of order $d_{n,k}=[x^n]\,g(x)f(x)^k$40, and makes transformations such as $d_{n,k}=[x^n]\,g(x)f(x)^k$41 representable by actual matrices (Prieto-Martínez et al., 10 Apr 2025).

A different analogue replaces power series by formal Dirichlet series

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

The Riordan-Dirichlet group uses multiplication matrices $d_{n,k}=[x^n]\,g(x)f(x)^k$43, composition-type matrices $d_{n,k}=[x^n]\,g(x)f(x)^k$44 built from

$d_{n,k}=[x^n]\,g(x)f(x)^k$45

and two-parameter matrices $d_{n,k}=[x^n]\,g(x)f(x)^k$46. Its group law mirrors the ordinary Riordan law: $d_{n,k}=[x^n]\,g(x)f(x)^k$47 and it supports a Dirichlet-series analogue of Lagrange inversion (Burlachenko, 2018).

Generalized Pascal matrices provide another extension. A $d_{n,k}=[x^n]\,g(x)f(x)^k$48-parametrized family interpolates between ordinary and exponential Riordan arrays: $d_{n,k}=[x^n]\,g(x)f(x)^k$49 while $d_{n,k}=[x^n]\,g(x)f(x)^k$50 produces arrays associated with $d_{n,k}=[x^n]\,g(x)f(x)^k$51-binomial coefficients. The exceptional case $d_{n,k}=[x^n]\,g(x)f(x)^k$52 forces a further enlargement to zero generalized Pascal matrices and zero generalized Riordan groups, where the usual coefficient series degenerates but a Riordan-like group law survives (Burlachenko, 2016).

The most recent three-series extension is the Sprugnoli group. Its elements $d_{n,k}=[x^n]\,g(x)f(x)^k$53 have columns

$d_{n,k}=[x^n]\,g(x)f(x)^k$54

with $d_{n,k}=[x^n]\,g(x)f(x)^k$55 odd. The fundamental theorem splits a series into even and odd bisections: $d_{n,k}=[x^n]\,g(x)f(x)^k$56 This group contains ordinary Riordan arrays when $d_{n,k}=[x^n]\,g(x)f(x)^k$57, contains double Riordan arrays under additional parity restrictions, and has a production matrix governed by two interlaced recurrence sequences $d_{n,k}=[x^n]\,g(x)f(x)^k$58 and $d_{n,k}=[x^n]\,g(x)f(x)^k$59 with $d_{n,k}=[x^n]\,g(x)f(x)^k$60 even (Barry, 15 May 2026).

Across these developments, Riordan matrices remain defined by the same structural principle: lower-triangular matrix representations of weighted substitution. The subsequent generalizations alter the coefficient kernel, the substitution algebra, the indexing set, or the parity pattern, but preserve the fundamental interaction between generating functions, formal composition, and matrix multiplication.