Jabotinsky matrix

In mathematics, a Jabotinsky matrix, or Bell 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.

Let ${\displaystyle f}$ be a formal power series. There exists coefficients ${\displaystyle (B_{n,k})_{n,k\geq 0}}$ such that$${\displaystyle f(x)^{k}=\sum _{n=0}^{\infty }B_{n,k}x^{n}.}$$The Jabotinsky matrix of ${\displaystyle f(x)}$ is defined as the infinite matrix^[1][2]

${\displaystyle \mathbf {B} (f)=\left({\begin{array}{cccc}B_{0,0}&B_{0,1}&B_{0,2}&\cdots \\B_{1,0}&B_{1,1}&B_{1,2}&\cdots \\B_{2,0}&B_{2,1}&B_{2,2}&\cdots \\\vdots &\vdots &\vdots &\ddots \end{array}}\right).}$

When ${\displaystyle f(0)=0}$, ${\displaystyle \mathbf {B} (f)}$ becomes an infinite lower triangular matrix whose the entries are given by ordinary Bell polynomials evaluated at the coefficients of ${\displaystyle f}$. This is why ${\displaystyle \mathbf {B} (f)}$ is oftentimes referred to as a Bell matrix.^[3][4]

Jabotinsky matrices have a long history, and were perhaps used for the first time in the context of iteration theory by Albert A. Bennett^[5] in 1915. Jabotinsky later pursued Bennett's research^[6][7][8] and applied them to Faber polynomials^[9]. Jabotinsky matrices were popularized during the 70s by Louis Comtet (fr)'s book Advanced Combinatorics, where he referred to them as iteration matrices, which is a denomination also sometimes used nowadays.^[10] This article's denomination appeared later^{[11][12][13][14][15]} and notably used by Donald Knuth.^[2]

Jabotinsky matrices satisfy the fundamental relationship$${\displaystyle {\textbf {B}}(f\circ g)={\textbf {B}}(g){\textbf {B}}(f)}$$

which makes the Jabotinsky matrix ${\displaystyle \mathbf {B} (f)}$ a (direct) representation of ${\displaystyle f(x)}$. Here the term ${\displaystyle f\circ g}$ denotes the composition of functions ${\displaystyle f(g(x))}$.

The fundamental property implies

• ${\displaystyle {\textbf {B}}(f^{n})={\textbf {B}}(f)^{n}}$, where ${\displaystyle f^{n}}$ is an iterated function and ${\displaystyle n}$ is a natural integer.
• ${\displaystyle {\textbf {B}}(f^{-1})={\textbf {B}}(f)^{-1}}$, where ${\displaystyle f^{-1}}$ is the inverse function, if ${\displaystyle f}$ has a compositional inverse.
• ${\displaystyle {\begin{bmatrix}1,x,x^{2},...\end{bmatrix}}{\textbf {B}}(f)={\begin{bmatrix}1,f(x),f(x)^{2},...\end{bmatrix}}.}$

Given a sequence ${\displaystyle (\Omega _{n})_{n\geq 0}}$, we can instead define the matrix with the coefficient ${\displaystyle (B_{n,k}^{\Omega })_{n,k\geq 0}}$ by^[1]$${\displaystyle \Omega _{k}f(x)^{k}=\sum _{n=0}^{\infty }B_{n,k}^{\Omega }\Omega _{n}x^{n}.}$$If ${\displaystyle (\Omega _{n})_{n\geq 0}}$ is the constant sequence equal to ${\displaystyle 1}$, we recover Jabotinsky matrices. In some contexts, the sequence is chosen to be ${\displaystyle \Omega _{n}=1/n!}$, so that the entry are given by regular Bell polynomials. This is a more convenient form for functions such as ${\displaystyle f(x)=-\log(1-x)}$ and ${\displaystyle f(x)=e^{x}-1}$ where Stirling numbers of the first and second kind appear in the matrices (see the examples).

This generalization gives a completely equivalent matrix since ${\displaystyle B_{n,k}^{\Omega }{\frac {\Omega _{n}}{\Omega _{k}}}=B_{n,k}}$.

• The Jabotinsky matrix of a constant is:

  ${\displaystyle \mathbf {B} (a)=\left({\begin{array}{cccc}1&0&0&\cdots \\a&0&0&\cdots \\a^{2}&0&0&\cdots \\\vdots &\vdots &\vdots &\ddots \end{array}}\right)}$

• The Jabotinsky matrix of a constant multiple is:

  ${\displaystyle {\textbf {B}}(cx)=\left({\begin{array}{cccc}1&0&0&\cdots \\0&c&0&\cdots \\0&0&c^{2}&\cdots \\\vdots &\vdots &\vdots &\ddots \end{array}}\right)}$

• The Jabotinsky matrix of the successor function:

  ${\displaystyle {\textbf {B}}(1+x)=\left({\begin{array}{ccccc}1&0&0&0&\cdots \\1&1&0&0&\cdots \\1&2&1&0&\cdots \\1&3&3&1&\cdots \\\vdots &\vdots &\vdots &\vdots &\ddots \end{array}}\right)}$

  The matrix displays Pascal's triangle.

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

  ${\displaystyle {\textbf {B}}(-\log(1-x))=\left({\begin{array}{cccccc}1&0&0&0&0&\cdots \\0&1&0&0&0&\cdots \\0&{\frac {1}{2}}&1&0&0&\cdots \\0&{\frac {1}{3}}&1&1&0&\cdots \\0&{\frac {1}{4}}&{\frac {11}{12}}&{\frac {3}{2}}&1&\cdots \\\vdots &\vdots &\vdots &\vdots &\vdots &\ddots \end{array}}\right)}$

  ${\displaystyle {\textbf {B}}(-\log(1-x))_{n,k}=\left{\frac {k!}{n!}}}$

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

  ${\displaystyle {\textbf {B}}(\exp(x)-1)=\left({\begin{array}{cccccc}1&0&0&0&0&\cdots \\0&1&0&0&0&\cdots \\0&{\frac {1}{2}}&1&0&0&\cdots \\0&{\frac {1}{6}}&1&1&0&\cdots \\0&{\frac {1}{24}}&{\frac {7}{12}}&{\frac {3}{2}}&1&\cdots \\\vdots &\vdots &\vdots &\vdots &\vdots &\ddots \end{array}}\right)}$

  ${\displaystyle {\textbf {B}}(\exp(x)-1)_{n,k}=\left\{{n \atop k}\right\}{\frac {k!}{n!}}}$

• The Jabotinsky matrix of exponential functions is given by ${\displaystyle {\textbf {B}}(\exp )_{n,k}={\frac {(ka)^{n}}{n!}}}$.

• Jabotinsky matrices are a special case of Riordan arrays.
• They are also related to Carleman linearization and Carleman (embedding) matrices^[16][17][18].

• Composition operator
• Schröder's equation
• Umbral calculus