Jabotinsky matrix

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

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

𝐁 (f) = (\begin{matrix} B_{0, 0} & B_{0, 1} & B_{0, 2} & \dots \\ B_{1, 0} & B_{1, 1} & B_{1, 2} & \dots \\ B_{2, 0} & B_{2, 1} & B_{2, 2} & \dots \\ ⋮ & ⋮ & ⋮ & ⋱ \end{matrix}) .

When $f (0) = 0$ , $𝐁 (f)$ becomes an infinite lower triangular matrix whose entries are given by ordinary Bell polynomials evaluated at the coefficients of $f$ . This is why $𝐁 (f)$ 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 Louis Comtet [fr]'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

Jabotinsky matrices satisfy the fundamental relationship $𝐁 (f \circ g) = 𝐁 (g) 𝐁 (f)$ which makes the Jabotinsky matrix $𝐁 (f)$ a (direct) representation of $f (x)$ . Here the term $f \circ g$ denotes the composition of functions $f (g (x))$ .

The fundamental property implies

$𝐁 (f^{n}) = 𝐁 (f)^{n}$ , where $f^{n}$ is an iterated function and $n$ is a natural number.
$𝐁 (f^{- 1}) = 𝐁 (f)^{- 1}$ , where $f^{- 1}$ is the inverse function, if $f$ has a compositional inverse.
$[\begin{matrix} 1, x, x^{2}, ... \end{matrix}] 𝐁 (f) = [\begin{matrix} 1, f (x), f (x)^{2}, ... \end{matrix}] .$

Generalization

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

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

Examples

The Jabotinsky matrix of a constant is:
$𝐁 (a) = (\begin{matrix} 1 & a & a^{2} & \dots \\ 0 & 0 & 0 & \dots \\ 0 & 0 & 0 & \dots \\ ⋮ & ⋮ & ⋮ & ⋱ \end{matrix})$
The Jabotinsky matrix of a constant multiple is:
$𝐁 (c x) = (\begin{matrix} 1 & 0 & 0 & \dots \\ 0 & c & 0 & \dots \\ 0 & 0 & c^{2} & \dots \\ ⋮ & ⋮ & ⋮ & ⋱ \end{matrix})$
The Jabotinsky matrix of the successor function:
$𝐁 (1 + x) = (\begin{matrix} 1 & 1 & 1 & 1 & \dots \\ 0 & 1 & 2 & 3 & \dots \\ 0 & 0 & 1 & 3 & \dots \\ 0 & 0 & 0 & 1 & \dots \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋱ \end{matrix})$

The matrix displays Pascal's triangle.
The Jabotinsky matrix of the exponential function is given by $𝐁 (\exp)_{n, k} = \frac{k^{n}}{n!}$ .
The Jabotinsky matrix of the logarithm is related to the (unsigned) Stirling numbers of the first kind scaled by factorials:
$𝐁 (- \log (1 - x)) = (\begin{matrix} 1 & 0 & 0 & 0 & 0 & \dots \\ 0 & 1 & 0 & 0 & 0 & \dots \\ 0 & \frac{1}{2} & 1 & 0 & 0 & \dots \\ 0 & \frac{1}{3} & 1 & 1 & 0 & \dots \\ 0 & \frac{1}{4} & \frac{11}{12} & \frac{3}{2} & 1 & \dots \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋱ \end{matrix})$

$𝐁 (- \log (1 - x))_{n, k} = [\binom{n}{k}] \frac{k!}{n!}$
The Jabotinsky matrix of the exponential function minus 1 is related to the Stirling numbers of the second kind scaled by factorials:
$𝐁 (\exp (x) - 1) = (\begin{matrix} 1 & 0 & 0 & 0 & 0 & \dots \\ 0 & 1 & 0 & 0 & 0 & \dots \\ 0 & \frac{1}{2} & 1 & 0 & 0 & \dots \\ 0 & \frac{1}{6} & 1 & 1 & 0 & \dots \\ 0 & \frac{1}{24} & \frac{7}{12} & \frac{3}{2} & 1 & \dots \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋱ \end{matrix})$

$𝐁 (\exp (x) - 1)_{n, k} = {\binom{n}{k}} \frac{k!}{n!}$

Alternative convention

Some authors, which include Bennett and Jabotinsky in their original work, use the transpose convention $𝐁 (f)^{T}$ for the matrix, which is an infinite upper triangular matrix when $f (0) = 0$ . The fundamental relationship becomes $𝐁 (f \circ g)^{T} = 𝐁 (f)^{T} 𝐁 (g)^{T},$ which 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.

Notes

1 2 Comtet, Louis (1974). Advanced Combinatorics: The Art of Finite and Infinite Expansions. Dordrecht: Springer Netherlands. ISBN 978-94-010-2198-2.
1 2 Knuth, D. (1992). "Convolution Polynomials". The Mathematica Journal. 2 (4): 67–78. arXiv:math/9207221. Bibcode:1992math......7221K.
↑ Aldrovandi, R.; Freitas, L. P. (1998-10-01). "Continuous iteration of dynamical maps". Journal of Mathematical Physics. 39 (10): 5324–5336. arXiv:physics/9712026. doi:10.1063/1.532574. ISSN 0022-2488.
↑ 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.
↑ Bennett, Albert A. (1915). "The Iteration of Functions of one Variable". The Annals of Mathematics. 17 (1): 23. doi:10.2307/2007213.
↑ 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.
↑ Jabotinsky, Eri (1955). "Iterational invariants". Technion. Israel Institute of Technology. Scientific Publications. 6: 64–80.
↑ Erdös, Paul; Jabotinsky, Eri (1960). "On analytic iteration". Journal d'Analyse Mathématique. 8 (1): 361–376. doi:10.1007/BF02786856. ISSN 0021-7670.
↑ 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.
1 2 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.
1 2 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. arXiv:1009.5518. doi:10.1016/j.jcta.2011.11.008.
↑ 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.
↑ 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.
↑ Lang, W. (2000). "On generalizations of the stirling number triangles". Journal of Integer Sequences. 3 (2.4): 1–19. Bibcode:2000JIntS...3...24L.
↑ 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.
↑ 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. arXiv:2105.05583. doi:10.1007/s00605-022-01687-0. ISSN 1436-5081.
1 2 Gralewicz, P.; Kowalski, K. (2002). "Continuous time evolution from iterated maps and Carleman linearization". Chaos, Solitons & Fractals. 14 (4): 563–572. arXiv:math-ph/0002044. doi:10.1016/S0960-0779(01)00222-3.
↑ Schippers, Eric (2006). "The Power Matrix, coadjoint action and quadratic differentials". Journal d'Analyse Mathématique. 98 (1): 249–277. doi:10.1007/BF02790277. ISSN 0021-7670.
↑ 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.
↑ Kowalski, Krzysztof; Steeb, W.-H. (1991). Nonlinear dynamical systems and Carleman linearization. Singapore ; Teaneck, N.J: World Scientific. ISBN 978-981-02-0587-4.

[:0-1] 1 2 Comtet, Louis (1974). Advanced Combinatorics: The Art of Finite and Infinite Expansions. Dordrecht: Springer Netherlands. ISBN 978-94-010-2198-2.

[:1-2] 1 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. arXiv:physics/9712026. 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 (1955). "Iterational invariants". Technion. Israel Institute of Technology. Scientific Publications. 6: 64–80.

[8] Erdös, Paul; Jabotinsky, Eri (1960). "On analytic iteration". Journal d'Analyse Mathématique. 8 (1): 361–376. doi:10.1007/BF02786856. ISSN 0021-7670.

[9] 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.

[:2-10] 1 2 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.

[:3-11] 1 2 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. arXiv:1009.5518. doi:10.1016/j.jcta.2011.11.008.

[12] 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.

[13] 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.

[14] Lang, W. (2000). "On generalizations of the stirling number triangles". Journal of Integer Sequences. 3 (2.4): 1–19. Bibcode:2000JIntS...3...24L.

[15] 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.

[16] 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. arXiv:2105.05583. doi:10.1007/s00605-022-01687-0. ISSN 1436-5081.

[:4-17] 1 2 Gralewicz, P.; Kowalski, K. (2002). "Continuous time evolution from iterated maps and Carleman linearization". Chaos, Solitons & Fractals. 14 (4): 563–572. arXiv:math-ph/0002044. doi:10.1016/S0960-0779(01)00222-3.

[18] Schippers, Eric (2006). "The Power Matrix, coadjoint action and quadratic differentials". Journal d'Analyse Mathématique. 98 (1): 249–277. doi:10.1007/BF02790277. ISSN 0021-7670.

[19] 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.

[20] Kowalski, Krzysztof; Steeb, W.-H. (1991). Nonlinear dynamical systems and Carleman linearization. Singapore ; Teaneck, N.J: World Scientific. ISBN 978-981-02-0587-4.

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