For a certain purpose, I want to treat $pb^n$ as a family of sequences over $n\in\mathbb{N}$ with different (fixed) values of $p$ and $b$. Ideally, I'd like to have a symbol $E : \mathbb{R}^2 \to \mathbb{R}^\mathbb{N}$, and an intelligible way to refer to individual sequences in that family and elements of those sequences.
Is there a standard way to write that?
Options I've considered:
- Since sequences are just functions whose domain is $\mathbb{N}$, I could use a notation for parameterized functions, such as $E\left(n;p,b\right)$ or $E\left(n|p,b\right)$ or $E_{p,b}\left(n\right)$; but since sequences are usually written with subscripts instead of parentheses, I'd rather put $n$ in a subscript, if possible. And except for $E_{p,b}\left(n\right)$ (which uses subscripts for exactly the wrong things), these don't seem to provide a way to refer to an individual sequence.
- I considered $E_n\left(p,b\right)$, which uses subscripts for the right thing and non-subscripts for the right things; but that seems to suggest a sequence of functions, $E : \mathbb{N} \to \mathbb{R}^{\left(\mathbb{R}^2\right)}$, rather than a family of sequences. (I suppose those concepts are isomorphic, but I want to be able to refer to specific sequences.)
- $E\left(p,b\right)_n$ or $\left(E\left(p,b\right)\right)_n$ has exactly the meaning I want, but it doesn't seem readable.
- I considered adapting a notation for parameterized functions to use subscripts instead of parentheses, such as $E_{n;p,b}$ or $E_{n|p,b}$ or $E_{\left(p,b\right)n}$, but I'm not sure if any of those is readable, either.
- I considered dropping the idea of denoting the family with a symbol, and just writing e.g. $\left(pb^n\right)_{n\in\mathbb{N}}$ or $\left\{pb^n\right\}_{n\in\mathbb{N}}$. This is what I'm leaning toward if there's no standard, or at least readable, way to use such a symbol.
Are any of the above options — or any other possibilities — standard? What's the best/clearest notation for something like this?
(If it's relevant, by the way — in my case, the parameters p and b will actually always be natural numbers. I wrote $\mathbb{R}$ rather than $\mathbb{N}$ above because their integer-ness isn't really relevant, and in order to help differentiate them from $n$; but if there's a notation that only makes sense for natural-valued parameters, I'm OK with that.)