Riordan array

A Riordan array is an <a href="/facts/Infinity/cF506uD7">infinite</a> lower <a href="/facts/Triangular_matrix/qjXSFMM3">triangular matrix</a>, 
 
 
 
 D
 
 
 {\displaystyle D}
 
, constructed from two <a href="/facts/Formal_power_series/CNmaG0Vk">formal power series</a>, 
 
 
 
 d
 (
 t
 )
 
 
 {\displaystyle d(t)}
 
 of order 0 and 
 
 
 
 h
 (
 t
 )
 
 
 {\displaystyle h(t)}
 
 of order 1, such that 
 
 
 
 
 d
 
 n
 ,
 k
 
 
 =
 [
 
 t
 
 n
 
 
 ]
 d
 (
 t
 )
 h
 (
 t
 
 )
 
 k
 
 
 
 
 {\displaystyle d_{n,k}=[t^{n}]d(t)h(t)^{k}}
 
.
A Riordan array is an element of the Riordan group. It was defined by mathematician <a href="/facts/Louis_Shapiro_(mathematician)/JkGkxTRg">Louis W. Shapiro</a> and named after <a href="/facts/John_Riordan_(mathematician)/kKrpSbPW">John Riordan</a>.
The study of Riordan arrays is a field influenced by and contributing to other areas such as <a href="/facts/Combinatorics/k14xUoKT">combinatorics</a>, <a href="/facts/Group_theory/NfowcI5H">group theory</a>, <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix theory</a>, <a href="/facts/Number_theory/zhoa7zrc">number theory</a>, <a href="/facts/Probability/fgYvMhwb">probability</a>, <a href="/facts/Sequence/gV2S4uqp">sequences</a> and <a href="/facts/Series_(mathematics)/yDnlsgIe">series</a>, <a href="/facts/Lie_group/a9jCQyjF">Lie groups</a> and <a href="/facts/Lie_algebra/n2gAUkNq">Lie algebras</a>, <a href="/facts/Orthogonal_polynomials/9cyRbDYU">orthogonal polynomials</a>, <a href="/facts/Graph_theory/VVPCd4TX">graph theory</a>, <a href="/facts/Network_theory/m6CQovwm">networks</a>, <a href="/facts/Unimodality/Xo3bhgeE">unimodal</a> sequences, combinatorial identities, <a href="/facts/Elliptic_curves/60jvaHSA">elliptic curves</a>, <a href="/facts/Numerical_analysis/QT9cYa9z">numerical approximation</a>, <a href="/facts/Asymptotics/kt87qLpu">asymptotic analysis</a>, and <a href="/facts/Data_analysis/CNtARxLW">data analysis</a>. Riordan arrays also unify tools such as <a href="/facts/Generating_functions/K7UVjnjL">generating functions</a>, <a href="/facts/Computer_algebra_system/jkLAC1U3">computer algebra systems</a>, <a href="/facts/Formal_language/crDTyP8q">formal languages</a>, and <a href="/facts/Path_analysis_(statistics)/8uCr2s27">path models</a>. Books on the subject, such as The Riordan Array (Shapiro et al., 1991), have been published.

Riordan array open-in-new

Riordan array