What are the properties of this set of infinite matrices and operations on them?

Question

Consider infinite matrices of the form

$$\left( \begin{array}{ccccc} a_0 & a_1 & a_2 & a_3 & . \\ 0 & a_0 & a_1 & a_2 & . \\ 0 & 0 & a_0 & a_1 & . \\ 0 & 0 & 0 & a_0 & . \\ . & . & . & . & . \\ \end{array} \right)$$

The elements on each diagonal coincide.

My questions are:

• Do they form a commutative ring?

• Can they be extended to form a field?

Now, let define an operation $\operatorname{reg} A=\sum_{k=0}^\infty B_k a_k,$

where $B_k$ are Bernoulli numbers.

What are the properties of this operation?

Let's define another operation $\det' A=\exp(\Re \operatorname{reg} \log A)$.

What are the properties of this operation?

Motivation part.

This is meant to be a matrix representation of divergent integrals and series. For instance,

$\sum_{k=1}^\infty 1= \left( \begin{array}{ccccc} 0 & 1 & 0 & 0 & . \\ 0 & 0 & 1 & 0 & . \\ 0 & 0 & 0 & 1 & . \\ 0 & 0 & 0 & 0 & . \\ . & . & . & . & . \\ \end{array} \right)$

$\sum_{k=0}^\infty 1= \left( \begin{array}{ccccc} 1 & 1 & 0 & 0 & . \\ 0 & 1 & 1 & 0 & . \\ 0 & 0 & 1 & 1 & . \\ 0 & 0 & 0 & 1 & . \\ . & . & . & . & . \\ \end{array} \right)$

$\sum_{k=0}^\infty k= \left( \begin{array}{ccccc} 1/12 & 1/2 & 1/2 & 0 & . \\ 0 & 1/12 & 1/2 & 1/2 & . \\ 0 & 0 & 1/12 & 1/2 & . \\ 0 & 0 & 0 & 1/12 & . \\ . & . & . & . & . \\ \end{array} \right)$

$\int_0^\infty x dx=\int_0^\infty \frac 2{x^3}=\left( \begin{array}{ccccc} 1/6 & 1/2 & 1/2 & 0 & . \\ 0 & 1/6 & 1/2 & 1/2 & . \\ 0 & 0 & 1/6 & 1/2 & . \\ 0 & 0 & 0 & 1/6 & . \\ . & . & . & . & . \\ \end{array} \right)$

There are also some expressions that include divergent integrals that can be represented this way:

$(-1)^{\int_0^\infty dx}=\left( \begin{array}{ccccccc} i & -\pi & -\frac{i \pi ^2}{2} & \frac{\pi ^3}{6} & \frac{i \pi ^4}{24} & -\frac{\pi ^5}{120} & . \\ 0 & i & -\pi & -\frac{i \pi ^2}{2} & \frac{\pi ^3}{6} & \frac{i \pi ^4}{24} & . \\ 0 & 0 & i & -\pi & -\frac{i \pi ^2}{2} & \frac{\pi ^3}{6} & . \\ 0 & 0 & 0 & i & -\pi & -\frac{i \pi ^2}{2} & . \\ 0 & 0 & 0 & 0 & i & -\pi & . \\ 0 & 0 & 0 & 0 & 0 & i & . \\ . & . & . & . & . & . & . \\ \end{array} \right)$

The $\operatorname{reg}$ operation gives the regularized value of the integral or series.

Answer 1

If the matrices have entries from a (unital) ring $R$ then the set of such matrices is isomorphic to $R[[x]]$, the ring of formal power series over $R$. To see this, observe that the map sending the infinite matrix with $a_0 = 0$, $a_1 = 1$ and $a_k = 0$ for $k \ge 2$ to $x$ is a ring isomorphism.

This also answers the second question: if $R$ is an integral domain then set of matrices embeds canonically in the field of fractions of $R[[x]]$ and this is the smallest field containing $R[[x]]$. In particular, if $R$ is a field then this field is $\{ \sum_{k=-m}^\infty a_k x^k : a_k \in R, m \in \mathbb{N}_0 \}$.

I'm uncertain how $\mathrm{reg}$ is (well)-defined, but certainly one can take $R$ to be the polynomial ring $\mathbb{C}[z]$ and then something like $\sum_{k=0}^\infty B_k(z) x^k$ is a well-defined element of $R[[x]] = \mathbb{C}[z][[x]]$. If, as in the correction then one wants Bernoulli numbers rather than the polynomials, just specialize to $\mathbb{C}[[x]]$ by evaluating at $z=0$.