Riordan array

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