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.



What are the properties of this set of infinite matrices and operations on them?
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 $z=0$.


