Jabotinsky matrix

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 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]}. Donald Knuth uses the name convolution matrix.^[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 the exponential function is given by ${\displaystyle {\textbf {B}}(\exp )_{n,k}={\frac {k^{n}}{n!}}}$.
• The Jabotinsky matrix of the logarithm is related to the (unsigned) Stirling numbers of the first kind scaled by factorials:

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

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

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

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

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

• 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