A Riordan array is an infinite lower triangular matrix,
[1] It was defined by mathematician Louis W. Shapiro and named after John Riordan.
[1] The study of Riordan arrays is a field influenced by and contributing to other areas such as combinatorics, group theory, matrix theory, number theory, probability, sequences and series, Lie groups and Lie algebras, orthogonal polynomials, graph theory, networks, unimodal sequences, combinatorial identities, elliptic curves, numerical approximation, asymptotic analysis, and data analysis.
Riordan arrays also unify tools such as generating functions, computer algebra systems, formal languages, and path models.
[2] Books on the subject, such as The Riordan Array[1] (Shapiro et al., 1991), have been published.
is the ring of formal power series with complex coefficients) is said to have order
for the set of formal power series of order
; it has a composition inverse that is there exists a power series
As mentioned previously, a Riordan array is usually defined via a pair of power series
The "array" part in its name stems from the fact that one associates to
of the array consists of the sequence of coefficients of the power series
is of order 0, it has a multiplicative inverse, and it follows that from the array's column 1 we can recover
is lower triangular and exhibits a geometric progression
It also follows that the map sending a pair of power series
It is not difficult to show that this pair generates the infinite triangular array of binomial coefficients
is a power series with associated coefficient sequence
Note that the matrix multiplication rules applied to infinite lower triangular matrices lead to finite sums only and the product of two infinite lower triangular matrices is infinite lower triangular.
The next two theorems were first stated and proved by Shapiro et al.[1], which describes them as derived from results in papers by Gian-Carlo Rota and the book of Roman.
be Riordan arrays, viewed as infinite lower triangular matrices.
Then the product of these matrices is the array associated to the pair
of formal power series, which is itself a Riordan array.
' of Riordan arrays viewed as pairs of power series by Proof: Since
which we can read as a linear combination of power series, namely
is the symbol for indicating correspondence on the power series level with matrix multiplication.
Theorem: The family of Riordan arrays endowed with the product '
[4] Theorem: An infinite lower triangular array
is a Riordan array if and only if there exist a sequence traditionally called the
Check that it is possible to solve the resulting equations for the coefficients of
be an infinite lower triangular array whose diagonal sequence
such that Proof: By the triangularity of the array, the equation claimed is equivalent to