The Carleman matrix of an infinitely differentiable function ${\displaystyle f(x)}$  is defined as:

${\displaystyle M[f]_{jk}={\frac {1}{k!}}\left[{\frac {d^{k}}{dx^{k}}}(f(x))^{j}\right]_{x=0}~,}$ 

so as to satisfy the (Taylor series) equation:

${\displaystyle (f(x))^{j}=\sum _{k=0}^{\infty }M[f]_{jk}x^{k}.}$ 

For instance, the computation of ${\displaystyle f(x)}$  by

${\displaystyle f(x)=\sum _{k=0}^{\infty }M[f]_{1,k}x^{k}.~}$ 

simply amounts to the dot-product of row 1 of ${\displaystyle M[f]}$  with a column vector ${\displaystyle \left[1,x,x^{2},x^{3},...\right]^{\tau }}$ .

The entries of ${\displaystyle M[f]}$  in the next row give the 2nd power of ${\displaystyle f(x)}$ :

${\displaystyle f(x)^{2}=\sum _{k=0}^{\infty }M[f]_{2,k}x^{k}~,}$ 

and also, in order to have the zeroth power of ${\displaystyle f(x)}$  in ${\displaystyle M[f]}$ , we adopt the row 0 containing zeros everywhere except the first position, such that

${\displaystyle f(x)^{0}=1=\sum _{k=0}^{\infty }M[f]_{0,k}x^{k}=1+\sum _{k=1}^{\infty }0\cdot x^{k}~.}$ 

Thus, the dot product of ${\displaystyle M[f]}$  with the column vector ${\displaystyle {\begin{bmatrix}1,x,x^{2},...\end{bmatrix}}^{T}}$  yields the column vector ${\displaystyle \left[1,f(x),f(x)^{2},...\right]^{T}}$ , i.e.,

${\displaystyle M[f]{\begin{bmatrix}1\\x\\x^{2}\\x^{3}\\\vdots \end{bmatrix}}={\begin{bmatrix}1\\f(x)\\(f(x))^{2}\\(f(x))^{3}\\\vdots \end{bmatrix}}.}$ 

A generalization of the Carleman matrix of a function can be defined around any point, such as:

${\displaystyle M[f]_{x_{0}}=M_{x}[x-x_{0}]M[f]M_{x}[x+x_{0}]}$ 

or ${\displaystyle M[f]_{x_{0}}=M[g]}$  where ${\displaystyle g(x)=f(x+x_{0})-x_{0}}$ . This allows the matrix power to be related as:

${\displaystyle (M[f]_{x_{0}})^{n}=M_{x}[x-x_{0}]M[f]^{n}M_{x}[x+x_{0}]}$ 

General Series edit

Another way to generalize it even further is think about a general series in the following way:

Let ${\displaystyle f(x)=\sum _{n}c_{n}(f)\cdot \psi _{n}(x)}$  be a series approximation of ${\displaystyle f(x)}$ , where ${\displaystyle \{\psi _{n}(x)\}_{n}}$  is a basis of the space containing ${\displaystyle f(x)}$ 

We can define ${\displaystyle G[f]_{mn}=c_{n}(\psi _{m}\circ f)}$ , therefore we have ${\displaystyle \psi _{m}\circ f=\sum _{n}c_{n}(\psi _{m}\circ f)\cdot \psi _{n}=\sum _{n}G[f]_{mn}\cdot \psi _{n}}$ , now we can prove that ${\displaystyle G[g\circ f]=G[g]\cdot G[f]}$ , if we assume that ${\displaystyle \{\psi _{n}(x)\}_{n}}$  is also a basis for ${\displaystyle g(x)}$  and ${\displaystyle g(f(x))}$ .

Let ${\displaystyle g(x)}$  be such that ${\displaystyle \psi _{l}\circ g=\sum _{m}G[g]_{lm}\cdot \psi _{m}}$  where ${\displaystyle G[g]_{lm}=c_{m}(\psi _{l}\circ g)}$ .

Now

${\displaystyle {\begin{aligned}\sum _{n}G[g\circ f]_{ln}\psi _{n}=\psi _{l}\circ (g\circ f)&=(\psi _{l}\circ g)\circ f\\&=\sum _{m}G[g]_{lm}(\psi _{m}\circ f)\\&=\sum _{m}G[g]_{lm}\sum _{n}G[f]_{mn}\psi _{n}\\&=\sum _{n,m}G[g]_{lm}G[f]_{mn}\psi _{n}\\&=\sum _{n}(\sum _{m}G[g]_{lm}G[f]_{mn})\psi _{n}\end{aligned}}}$ 

Comparing the first and the last term, and from ${\displaystyle \{\psi _{n}(x)\}_{n}}$  being a base for ${\displaystyle f(x)}$ , ${\displaystyle g(x)}$  and ${\displaystyle g(f(x))}$  it follows that ${\displaystyle G[g\circ f]=\sum _{m}G[g]_{lm}G[f]_{mn}=G[g]\cdot G[f]}$ 

Examples edit

If we set ${\displaystyle \psi _{n}(x)=x^{n}}$  we have the Carleman matrix

If ${\displaystyle \{e_{n}(x)\}_{n}}$  is an orthonormal basis for a Hilbert Space with a defined inner product ${\displaystyle \langle f,g\rangle }$ , we can set ${\displaystyle \psi _{n}=e_{n}}$  and ${\displaystyle c_{n}(f)}$  will be ${\displaystyle {\displaystyle \langle f,e_{n}\rangle }}$ . If ${\displaystyle e_{n}(x)=e^{inx}}$  we have the analogous for Fourier Series, namely ${\displaystyle c_{n}(f)={\cfrac {1}{2\pi }}\int _{-\pi }^{\pi }\displaystyle f(x)\cdot e^{-inx}dx}$ 

Carleman matrices satisfy the fundamental relationship

• ${\displaystyle M[f\circ g]=M[f]M[g]~,}$ 

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

Other properties include:

• ${\displaystyle \,M[f^{n}]=M[f]^{n}}$ , where ${\displaystyle \,f^{n}}$  is an iterated function and
• ${\displaystyle \,M[f^{-1}]=M[f]^{-1}}$ , where ${\displaystyle \,f^{-1}}$  is the inverse function (if the Carleman matrix is invertible).

The Carleman matrix of a constant is:

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

The Carleman matrix of the identity function is:

${\displaystyle M_{x}[x]=\left({\begin{array}{cccc}1&0&0&\cdots \\0&1&0&\cdots \\0&0&1&\cdots \\\vdots &\vdots &\vdots &\ddots \end{array}}\right)}$ 

The Carleman matrix of a constant addition is:

${\displaystyle M_{x}[a+x]=\left({\begin{array}{cccc}1&0&0&\cdots \\a&1&0&\cdots \\a^{2}&2a&1&\cdots \\\vdots &\vdots &\vdots &\ddots \end{array}}\right)}$ 

The Carleman matrix of the successor function is equivalent to the Binomial coefficient:

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

${\displaystyle M_{x}[1+x]_{jk}={\binom {j}{k}}}$ 

The Carleman matrix of the logarithm is related to the (signed) Stirling numbers of the first kind scaled by factorials:

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

${\displaystyle M_{x}[\log(1+x)]_{jk}=s(k,j){\frac {j!}{k!}}}$ 

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

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

${\displaystyle M_{x}[-\log(1-x)]_{jk}=|s(k,j)|{\frac {j!}{k!}}}$ 

The Carleman matrix of the exponential function is related to the Stirling numbers of the second kind scaled by factorials:

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

${\displaystyle M_{x}[\exp(x)-1]_{jk}=S(k,j){\frac {j!}{k!}}}$ 

The Carleman matrix of exponential functions is:

${\displaystyle M_{x}[\exp(ax)]=\left({\begin{array}{ccccc}1&0&0&0&\cdots \\1&a&{\frac {a^{2}}{2}}&{\frac {a^{3}}{6}}&\cdots \\1&2a&2a^{2}&{\frac {4a^{3}}{3}}&\cdots \\1&3a&{\frac {9a^{2}}{2}}&{\frac {9a^{3}}{2}}&\cdots \\\vdots &\vdots &\vdots &\vdots &\ddots \end{array}}\right)}$ 

${\displaystyle M_{x}[\exp(ax)]_{jk}={\frac {(ja)^{k}}{k!}}}$ 

The Carleman matrix of a constant multiple is:

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

The Carleman matrix of a linear function is:

${\displaystyle M_{x}[a+cx]=\left({\begin{array}{cccc}1&0&0&\cdots \\a&c&0&\cdots \\a^{2}&2ac&c^{2}&\cdots \\\vdots &\vdots &\vdots &\ddots \end{array}}\right)}$ 

The Carleman matrix of a function ${\displaystyle f(x)=\sum _{k=1}^{\infty }f_{k}x^{k}}$  is:

${\displaystyle M[f]=\left({\begin{array}{cccc}1&0&0&\cdots \\0&f_{1}&f_{2}&\cdots \\0&0&f_{1}^{2}&\cdots \\\vdots &\vdots &\vdots &\ddots \end{array}}\right)}$ 

The Carleman matrix of a function ${\displaystyle f(x)=\sum _{k=0}^{\infty }f_{k}x^{k}}$  is:

${\displaystyle M[f]=\left({\begin{array}{cccc}1&0&0&\cdots \\f_{0}&f_{1}&f_{2}&\cdots \\f_{0}^{2}&2f_{0}f_{1}&f_{1}^{2}+2f_{0}f_{2}&\cdots \\\vdots &\vdots &\vdots &\ddots \end{array}}\right)}$ 

The Bell matrix or the Jabotinsky matrix of a function ${\displaystyle f(x)}$  is defined as^[1][2][3]

${\displaystyle B[f]_{jk}={\frac {1}{j!}}\left[{\frac {d^{j}}{dx^{j}}}(f(x))^{k}\right]_{x=0}~,}$ 

so as to satisfy the equation

${\displaystyle (f(x))^{k}=\sum _{j=0}^{\infty }B[f]_{jk}x^{j}~,}$ 

These matrices were developed in 1947 by Eri Jabotinsky to represent convolutions of polynomials.^[4] It is the transpose of the Carleman matrix and satisfy

${\displaystyle B[f\circ g]=B[g]B[f]~,}$  which makes the Bell matrix B an anti-representation of ${\displaystyle f(x)}$ .

• Carleman linearization
• Composition operator
• Function composition
• Schröder's equation
• Bell polynomials

• R Aldrovandi, Special Matrices of Mathematical Physics: Stochastic, Circulant and Bell Matrices, World Scientific, 2001. (preview)
• R. Aldrovandi, L. P. Freitas, Continuous Iteration of Dynamical Maps, online preprint, 1997.
• P. Gralewicz, K. Kowalski, Continuous time evolution from iterated maps and Carleman linearization, online preprint, 2000.
• K Kowalski and W-H Steeb, Nonlinear Dynamical Systems and Carleman Linearization, World Scientific, 1991. (preview)