In mathematics, the Jabotinsky matrix (sometimes called iteration matrix or power matrix) is an infinite 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

be a formal power seriesThe Jabotinsky matrix is defined as the infinite matrix

becomes an infinite lower triangular matrix whose entries are given by ordinary Bell polynomials is sometimes 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 (which is a denomination also sometimes used nowadays). This article's denomination appeared later. Donald Knuth uses the name convolution matrix.

Properties

denotes the composition.

The fundamental property implies

• is an iterated function is a natural number.

• is the inverse function has a compositional inverse.

Generalization

by, so that the entry are given by regular Bell polynomials where Stirling numbers of the first and second kind appear in the matrices (see the examples).

.

Examples

• The Jabotinsky matrix of a constant is:

• The Jabotinsky matrix of a constant multiple is:

• The Jabotinsky matrix of the successor function:

  1. The matrix displays Pascal's triangle.

  The Jabotinsky matrix of the exponential function.

• The Jabotinsky matrix of the logarithm is related to the (unsigned) Stirling numbers of the first kind scaled by factorials:

• The Jabotinsky matrix of the exponential function minus 1 is related to the Stirling numbers of the second kind scaled by factorials:

Alternative convention

Some authors, which include Bennett and Jabotinsky in their original work, use the transpose for the matrix, which is an infinite upper triangular matrixwhich preserves the order of the operations.

Related matrices

• Jabotinsky matrices are a special case of the later introduced Riordan arrays.

• They are also related to Carleman linearization and Carleman (embedding) matrices.

• Grunsky matrices can be expressed with Jabotinsky matrices.

See also

• Composition operator

• Schröder's equation

• Faà di Bruno's formula

• Umbral calculus

Notes