# Notation for meta-variable or placeholder variable

I have an ODE model for a biological system which has state variables named $$A, B, C, \dots$$, and the expressions for their derivatives contain the unknown functions $$f_A, f_B, f_C, \dots$$.

I was typing up a description for each but it ended being very repetitive:

$$\begin{matrix} f_A :& (\text{function related to variable } A )\\ f_B :& (\text{function related to variable } B )\\ f_C :& (\text{function related to variable } C )\\ \vdots & \vdots \end{matrix}$$

If I had labeled my variables $$x_1, x_2, x_3, \dots$$ it would be a lot more clear to write

$$f_i : \; ( \text{function related to variable } x_i ), \quad i = 1 \dots n$$

In this case $$i$$ is the common notation for an integer index. Is there an equivalent standard notation when the value of the variable is a symbol, or another variable? I've been using the $$*$$ symbol:

$$f_* : \quad ( \text{function related to variable } * ), \quad * \in \{A, B, C, \dots\}$$

but it seems confusing. The important thing is that the domain of $$*$$ is the set of variables $$\{A, B, C, \dots\}$$, not the set of values of those variables (which are real numbers). This is why I refer to it as a meta-variable in the title, although I'm not sure that is the correct term.

I've seen $$\square$$ used a as a placeholder for a binary operator, like $$x \square y$$, don't know if that applies here. All information I've been able to find is specifically for mathematical logic.

## 1 Answer

What you seem to be describing is an indexed family of functions, where the indexing set is something other than the natural numbers. In my own work, I have used the notation $$\{ f_{\imath} \}_{\imath \in \mathscr{I} }.$$ Here, $$\imath$$ (typset as \imath) is an index and $$\mathscr{I}$$ (typeset as \mathscr{I}) is an index set, which could be anything (the natural numbers, the real numbers, the $$p$$-adic numbers, a collection of variables (whatever that means), the ingredients in my pancake recipe... whatevs). Folland uses a similar notation in his text on real analysis. Specifically, in his section on topology, he uses $$\langle f_{\lambda} \rangle$$ to denote a net (a kind of generalized sequence), where the index $$\lambda$$ is contained in some index set $$\Lambda$$.

The overall point here is that the specific symbols here aren't really that important—the important idea is to recognize that any set can serve as an index set. As long as you clearly indicate what your index set is, there is likely to be no confusion.

That being said, I would probably avoid something like $$f_\ast$$. The $$\ast$$ looks like an unspecified index to me, which implies that you could plug in just about any ol' thing. It lacks specificity (though that is a purely opinionated aesthetic judgement). It is also possible, depending on context, that this could be confused with other uses of $$(\cdot)_{\ast}$$ or $$(\cdot)^\ast$$, such as (co)homology groups or the adjoint operator.

Finally (and, again, this is an opinion), taking your index set to be $$\mathscr{I}$$ or $$\Lambda$$ makes it easier to write down distinct elements. For example, $$\iota$$ and $$\jmath$$ (or even $$i$$ and $$j$$) "look like" elements of an index set $$\mathscr{I}$$, while $$\lambda$$, $$\mu$$, $$\nu$$ (and so on) "look like" elements of an index set $$\Lambda$$. To what set does $$\ast$$ belong, and what are the other elements of that set? †, §, and so on?

• Thanks Xander, I am indeed describing an indexed family of functions. I've edited the question to make the domains of i and * more clear. I think a big part of my confusion is that * is a variable who's value is a reference to another variable {A, B, C, ...} (or its symbol), not the value of the other variable, if that makes sense. Unfortunately greek letters don't work well here because I am using them for parameters. – JaredL Feb 26 at 5:01
• As I said in my answer above, you can use $\ast$ as an index from the set $\{A, B, C,\dotsc\}$, but such notation would be unusual and might cause confusion. I think that you are better off using (for example) $\lambda$, $\mu$, $\nu$, etc as indices from the set $\Lambda = \{A, B, C, \dotsc\}$. – Xander Henderson Feb 26 at 5:05