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.

Definition

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, and 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]} .

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 ^[5] in 1915. Jabotinsky later pursued Bennett's research ^{[6] [7]} and applied them to Faber polynomials ^[8] , after Issai Schur rediscovered Jabotinsky matrices in about 1940 ^[9] while working on Faber polynomials. 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] .

Properties

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.
• ${\displaystyle {\begin{bmatrix}1,x,x^{2},...\end{bmatrix}}{\textbf {B}}(f)={\begin{bmatrix}1,f(x),f(x)^{2},...\end{bmatrix}}.}$

Generalization

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 form is more convenient for the functions ${\displaystyle f(x)=-\log(1-x)}$ 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 ${\displaystyle B_{n,k}^{\Omega }{\frac {\Omega _{n}}{\Omega _{k}}}=B_{n,k}}$.

Examples

• 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:

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

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

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

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

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

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

Related matrices

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

See also

• Composition operator
• Schröder's equation
• Umbral calculus

Notes

1. Comtet, Louis (1974). Advanced Combinatorics: The Art of Finite and Infinite Expansions. Dordrecht: Springer Netherlands. ISBN   978-94-010-2198-2.
2. Knuth, D. (1992). "Convolution Polynomials". The Mathematica Journal. 2 (4): 67–78. arXiv: math/9207221 . Bibcode:1992math......7221K.
3. Aldrovandi, R.; Freitas, L. P. (1998-10-01). "Continuous iteration of dynamical maps". Journal of Mathematical Physics. 39 (10): 5324–5336. doi:10.1063/1.532574. ISSN   0022-2488.
4. Aldrovandi, R. (2001). Special matrices of mathematical physics: stochastic, circulant, and Bell matrices. Singapore ; River Edge, N.J: World Scientific. ISBN   978-981-02-4708-9.
5. Bennett, Albert A. (1915). "The Iteration of Functions of one Variable". The Annals of Mathematics. 17 (1): 23. doi:10.2307/2007213.
6. Jabotinsky, Eri (1947). "Sur la représentation de la composition de fonctions par un produit de matrices. Applicaton à l'itération de e^x et de e^x-1". Comptes rendus de l'Académie des Sciences. 224: 323–324.
7. Jabotinsky, Eri (1963). "Analytic iteration". Transactions of the American Mathematical Society. 108 (3): 457–477. doi:10.1090/S0002-9947-1963-0155971-X. ISSN   0002-9947.
8. Jabotinsky, Eri (1953). "Representation of functions by matrices. Application to Faber polynomials" . Proceedings of the American Mathematical Society. 4 (4): 546–553. doi: 10.1090/S0002-9939-1953-0059359-0 . ISSN   0002-9939.
9. Schur, Issai (1945). "On Faber Polynomials". American Journal of Mathematics. 67 (1): 33. doi:10.2307/2371913.
10. Aschenbrenner, Matthias (2012). "Logarithms of iteration matrices, and proof of a conjecture by Shadrin and Zvonkine". Journal of Combinatorial Theory, Series A. 119 (3): 627–654. doi:10.1016/j.jcta.2011.11.008.
11. Lavoie, J. L.; Tremblay, R. (1981). "The Jabotinsky Matrix of a Power Series". SIAM Journal on Mathematical Analysis. 12 (6): 819–825. doi:10.1137/0512067. ISSN   0036-1410.
12. Brini, Andrea (1984-05-01). "Higher dimensional recursive matrices and diagonal delta sets of series". Journal of Combinatorial Theory, Series A. 36 (3): 315–331. doi:10.1016/0097-3165(84)90039-6. ISSN   0097-3165.
13. Lang, W. (2000). "On generalizations of the stirling number triangles". Journal of Integer Sequences. 3 (2.4): 1–19. Bibcode:2000JIntS...3...24L.
14. Mansour, Toufik; Schork, Matthias; Shattuck, Mark (2012-11-01). "On the Stirling numbers associated with the meromorphic Weyl algebra". Applied Mathematics Letters. 25 (11): 1767–1771. doi:10.1016/j.aml.2012.02.009. ISSN   0893-9659.
15. Sokal, Alan D. (2023-02-01). "Total positivity of some polynomial matrices that enumerate labeled trees and forests I: forests of rooted labeled trees". Monatshefte für Mathematik. 200 (2): 389–452. doi:10.1007/s00605-022-01687-0. ISSN   1436-5081.
16. Tsiligiannis, C. A; Lyberatos, G (1987-08-15). "Steady state bifurcations and exact multiplicity conditions via Carleman linearization". Journal of Mathematical Analysis and Applications. 126 (1): 143–160. doi:10.1016/0022-247X(87)90082-5. ISSN   0022-247X.
17. Kowalski, Krzysztof; Steeb, W.-H. (1991). Nonlinear dynamical systems and Carleman linearization. Singapore ; Teaneck, N.J: World Scientific. ISBN   978-981-02-0587-4.
18. Gralewicz, P.; Kowalski, K. (2002). "Continuous time evolution from iterated maps and Carleman linearization". Chaos, Solitons & Fractals. 14 (4): 563–572. doi:10.1016/S0960-0779(01)00222-3.