Recently I learned about a matrix called Carleman matrix. It is a matrix used to represent function iteration with matrix multiplying.

Carleman linearization is a technique used to embed a finite dimensional system of analytic ordinary differential equations into an infinite system of linear differential equations[2]

I noticed that carleman matrix may be a generalization of Maclaurin series.

- I researched about Maclaurin series of multivariable function.

$$a_{kl}=\frac 1{(l-k)!k!} \left [ \frac {\partial^l f(x,y)}{\partial x^k\partial y^{l-k}} \right ]_{ x=0, y=0}$$

$$f(x,y)=\sum^\infty_{l=0}\sum^l_{k=0}a_{kl}x^ky^{l-k}$$

And I modified above to make it look like carleman matrix:

$$a_{jkl}=\frac 1{(l-k)!k!} \left [ \frac {\partial^l f(x,y)^j}{\partial x^k\partial y^{l-k}} \right ]_{ x=0, y=0}$$

$$f(x,y)^j=\sum^\infty_{l=0}\sum^l_{k=0}a_{jkl}x^ky^{l-k}$$.

Which gives me something like a **Carleman tensor. (I COINED IT)**

Is my modification correct? And if then, could you teach me how to apply this tensor?

E.g.

How can I create a carleman tensor with multivariable functions such as $f(x,y)=x+y$?

(Like the Maclaurin series of a multivariate function).

**Example of desired answers**:

The carleman coefficients for $f(x,y)=x+y$ are:

$$M_{xy}[x+y]_{jk}= \cdots$$

**Also** I want to know how to iterate functions with the multivariable carleman tensors like:

$M[f(g(x),h(y))]=M[f]\bigoplus$ Concatenate $\left(M[g], M[h] \right)$

Which might *not* (mostly) be correct.

**Edit #2**

I think I need to find out how multivariable functions can be **coordinatizated** in orthogonal function space.

**Edit #3**

I suddenly came up with an idea, namely,

Partial carlemann matrix

And

Total carlemann matrix $:=\sum \bigoplus \text{partial carlemann matrix}$.

Hope that the above idea helps as a clue..