# Generalization of Carleman coefficients to multivariable functions - Carleman tensor?

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

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).

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..

• I found some useful material here: sciencedirect.com/science/article/pii/S1474667017313149/… Aug 31 '18 at 15:13
• Please be free to comment on this. I'll be fully appreciated with your small comments as I am not an authority nor an expert(I'm just a high schooler). Are my expressions about multivirate maclaurin series correct?(I've just guessed from this answer ( math.stackexchange.com/a/1020025/553404) and might be incorrect) Sep 1 '18 at 9:12
• There was a critical error on this maclaurin series and I fixed it.. Sep 4 '18 at 14:38
• You have made many edits to this question during the last 7 days, which suggests that the question is not clearly formed (or that you are making minor edits needlessly) Sep 7 '18 at 14:05
• Just found one downloaded article "Algebras with ternary law of composition and their realization by cubic matrices" by V. Abramov & al on arxiv. (0901.2506v1) Jan 2009. But there are older articles and small books about that subject as far as I remember. Oct 10 '18 at 9:13