Let $q$ be a prime power and $\mathbb{F}_q$ the field of cardinality $q$. Let $A = \mathbb{F}_q[T]$ and let $A_+ \subset A$ be the monic polynomials. Choose any ordering $<$ of $A_+$ and let $k$ be a positive integer. Set $$e(k) = \sum_{\begin{matrix} a_1, a_2, \ldots, a_k \in A_+ \\ a_1<a_2<\cdots<a_k \end{matrix}} \frac{1}{a_1 a_2 \cdots a_k}$$ where the sum is considered in the $T^{-1}$-adic topology. In non-archimedean analysis there is no need to discuss the order of summation.

I'm trying to find out what is known about algebraic relations between the $e(k)$ over $\mathbb{F}_q(T)$. Here is what I have found so far:

The $e(k)$ are very close to the Taylor coefficients of Thakur's geometric $\Gamma$ function: $$\tfrac{1}{\Gamma(z)} = z \prod_{a \in A_+} (1+z/a).$$ (See Thakur, 1991, Section 5.) But the papers I found in a quick search concentrate on evaluating $\Gamma$ at points of $\mathbb{F}_q(T)$, not the coefficients of the Taylor series.

The sum $\zeta(k) = \sum_{a \in A_+} \tfrac{1}{a^k}$ is in the ring generated by the $e(k)$. This sum is called the Goss $\zeta$-function and is very well understood: We have the relations $\zeta(kp) = \zeta(k)^p$ and $\zeta((q-1) m) = B(m) \pi^{(q-1)m}$ (for positive integer $m$) where $B(m)$ is an element of $\mathbb{F}_q(t)$ analogous to a Bernoulli number and $\pi$ is a transcendental element of $\mathbb{F}_q[[(-T)^{-1/(q-1)}]]$. Other than these relations, the $\zeta(m)$ are algebraically independent over $\mathbb{F}_q(T)$. See Yu, 1991.

Newton's identities show that the ring generated by the $e(k)$ is the same as the ring generated by the $\zeta(k)$ and the $e(pk)$. Obviously, this leaves room for the ring generated by the $e(k)$ to be much larger.

Is the ring generated by the $e(k)$ actually this much larger, or am I missing a trick?

Chang 2012 and others study $$\zeta(s_1, s_2, \ldots, s_k):= \sum_{\deg(a_1) < \deg(a_2) < \cdots < \deg(a_k)} \frac{1}{a_1^{s_1} \cdots a_k^{s_k}}.$$

Is this version of multiple $\zeta$ values related to my question, or is it unrelated?

If we remove the monic condition in the summation, then the sum is of the form $\pi^k Q(k)$ for some $Q(k) \in \mathbb{F}_q(T)$. This is Proposition 2.1 in this note of mine.

The reason I started going into this question is that I am spending today adding references to this paper in preparation to putting it on the arXiv and sending it off.