Jabotinsky matrix


In mathematics, the Jabotinsky matrix is a matrix used to convert function composition into matrix multiplication. It is often used in iteration theory to find the continuous iteration of functions. The matrix is named after mathematician Eri Jabotinsky.

Definition

Let be a formal power series. There exists coefficients such thatThe Jabotinsky matrix of is defined as the infinite matrix
When, becomes an infinite lower triangular matrix whose entries are given by ordinary Bell polynomials evaluated at the coefficients of. This is why is oftentimes referred to as a Bell matrix.

History

Jabotinsky matrices have a long history, and were perhaps used for the first time in the context of iteration theory by Albert A. Bennett in 1915. Jabotinsky later pursued Bennett's research and applied them to Faber polynomials. Jabotinsky matrices were popularized during the 70s by 's book Advanced Combinatorics, where he referred to them as iteration matrices. This article's denomination appeared later. Donald Knuth uses the name convolution matrix.

Properties

Jabotinsky matrices satisfy the fundamental relationship
which makes the Jabotinsky matrix a representation of. Here the term denotes the composition of functions.
The fundamental property implies
Given a sequence, we can instead define the matrix with the coefficient byIf is the constant sequence equal to, we recover Jabotinsky matrices. In some contexts, the sequence is chosen to be, so that the entry are given by regular Bell polynomials. This is a more convenient form for functions such as and where Stirling numbers of the first and second kind appear in the matrices.
This generalization gives a completely equivalent matrix since.

Examples