Jabotinsky matrix

Definition

Let f be a formal power series. There exists coefficients (B_{n,k}){n, k\geq 0} such thatf(x)^k = \sum{n=0}^{\infty} B_{n,k} x^n.The Jabotinsky matrix of f(x) is defined as the infinite matrix :\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 f(0) = 0, \mathbf 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 \mathbf B(f) is oftentimes 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

Jabotinsky matrices satisfy the fundamental relationship\textbf B(f \circ g) = \textbf B(g)\textbf B(f)

which makes the Jabotinsky matrix \mathbf 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

• \textbf B(f^n) = \textbf B(f)^n, where f^n is an iterated function and n is a natural integer.
• \textbf B(f^{-1}) = \textbf B(f)^{-1}, where f^{-1} is the inverse function, if f has a compositional inverse.
• \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 (\Omega_n){n\ge0}, we can instead define the matrix with the coefficient (B{n,k}^\Omega){n, k\geq 0} by\Omega_k f(x)^k = \sum{n=0}^{\infty} B^\Omega_{n,k} \Omega_n x^n.If (\Omega_n)_{n\ge0} is the constant sequence equal to 1, we recover Jabotinsky matrices. In some contexts, the sequence is chosen to be \Omega_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}^\Omega \frac{\Omega_n}{\Omega_k} = B_{n,k}.

Examples

• The Jabotinsky matrix of a constant is:
• :\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:
• :\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:
• :\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 \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:
• :\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)
• :\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:
• :\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)
• :\textbf B(\exp(x) - 1)_{n,k} = \left{{n \atop k}\right} \frac{k!}{n!}

Related matrices

• Jabotinsky matrices are a special case of Riordan arrays.
• They are also related to Carleman linearization and Carleman (embedding) matrices*.*

Notes

References

1. Comtet, Louis. (1974). "Advanced Combinatorics: The Art of Finite and Infinite Expansions". Springer Netherlands.
2. Knuth, D.. (1992). "Convolution Polynomials". The Mathematica Journal.
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4. Aldrovandi, R.. (2001). "Special matrices of mathematical physics: stochastic, circulant, and Bell matrices". World Scientific.
5. Bennett, Albert A.. (1915). "The Iteration of Functions of one Variable". The Annals of Mathematics.
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.
7. Erdös, Paul. (1960). "On analytic iteration". Journal d'Analyse Mathématique.
8. Jabotinsky, Eri. (1963). "Analytic iteration". Transactions of the American Mathematical Society.
9. Jabotinsky, Eri. (1953). "Representation of functions by matrices. Application to Faber polynomials". Proceedings of the American Mathematical Society.
10. Aschenbrenner, Matthias. (2012). "Logarithms of iteration matrices, and proof of a conjecture by Shadrin and Zvonkine". Journal of Combinatorial Theory, Series A.
11. Lavoie, J. L.. (1981). "The Jabotinsky Matrix of a Power Series". SIAM Journal on Mathematical Analysis.
12. Brini, Andrea. (1984-05-01). "Higher dimensional recursive matrices and diagonal delta sets of series". Journal of Combinatorial Theory, Series A.
13. Lang, W.. (2000). "On generalizations of the stirling number triangles". Journal of Integer Sequences.
14. Mansour, Toufik. (2012-11-01). "On the Stirling numbers associated with the meromorphic Weyl algebra". Applied Mathematics Letters.
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.
16. Tsiligiannis, C. A. (1987-08-15). "Steady state bifurcations and exact multiplicity conditions via Carleman linearization". Journal of Mathematical Analysis and Applications.
17. Kowalski, Krzysztof. (1991). "Nonlinear dynamical systems and Carleman linearization". World Scientific.
18. Gralewicz, P.. (2002). "Continuous time evolution from iterated maps and Carleman linearization". Chaos, Solitons & Fractals.