Consider a multiple series of the form: $$ \sum_{k_1=0}^\infty\cdots\sum_{k_n=0}^\infty f(k_1,\dots,k_n). $$ I have a more complicated multiple series where the indicies are $k_{jn}$ with $j=1,\dots,J$ and $n=1,\dots,n_j$ (note that $n_j$ changes with $j$). How can I compactly represent such a series?
Perhaps: $$ \sum_{\mathbf k=0}^\infty f(\mathbf k), $$ where $\mathbf k=(k_{1,1},\dots,k_{1,n_1},\dots,k_{J,1},\dots,k_{J,n_J})$? I feel like this too is very clunky. Is there a generally accepted notation for writing such series?
You can use multi-index, (it can be shaped in many different ways in order to have more various sums)
$$\sum_{\alpha \in A} f(\alpha)$$ with $A \subset \mathbb{N}^{(\mathbb{N})}$ (finished support sequences or "almost null sequences") describing your stuff (it is ok since it is countable). But maybe this is not what you want?
I would "put" $k_1,\dots,k_{n_j}$ in $A$ as the sequence $(k_1,\dots,k_{n_j}, 0, \dots , 0, \dots)$
However, if you want to be able to have any integer (including $0$), we could define $Z = \mathbb{Z} \sqcup \{\ast\}$ where $\ast$ will replace your zero, take $A \subset Z^{(\mathbb{N})}$ is given by the sequence where for some $N$ you have $u_k = \ast$ for $k > N$. Then $(k_1,\dots,k_{n_j})$ would be encoded as $(k_1,\dots,k_{n_j}, \ast, \dots , \ast, \dots)$
The only thing you need to have is the summability of your family $f(\alpha)_\alpha$.
I have another Idea with a double sum: $$\sum_{i = 1}^J \sum_{k \in \mathbb{N}^{n_j}} f_i(k)$$
but the problem here is that $f_i$ lives in a different space each time. You could consider the filtered set: $$N_\infty = \bigsqcup_{i=0}^\infty \mathbb{N}^i$$ which is countable and define $f$ on $A \subset N_\infty$ then you just have: $$\sum_{\alpha \in A} f(\alpha) $$
Anyway be careful with the definition of $f$