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

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.

• Could you add some motivation for the parts of the question on Bernoulli polynomials? Sep 21, 2021 at 23:10
• @MarkWildon yes, just a moment. Sep 21, 2021 at 23:11
• @MarkWildon done. By the way, a typo: it's Bernoulli numbers. Sep 21, 2021 at 23:32
• Thanks for the examples, but I still don't understand the motivation. How does one use Bernoulli numbers to express the definite integral as a matrix / formal power series? Could you give an explicit example of the summation? Is it using Euler's summation formula? Sep 21, 2021 at 23:49
• @MarkWildon Bernoulli numbers are used for regularization. The integrals can be transformed using these formulas: mathoverflow.net/questions/403704/… If you have a MathML-enabled browser (Firefox-based), here is my wiki: exnumbers.miraheze.org/wiki/… Also, look here: mathoverflow.net/questions/115743/an-algebra-of-integrals/… Sep 21, 2021 at 23:54

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 $$z=0$$.