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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.)

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  • $\begingroup$ I’d say both $E(p,b)_n$ and putting everything in subscripts are fine. For the latter, if you want to think about these primarily as sequences, you should put $n$ last, i.e. $E_{p,b,n}$ or even $E_{pbn}$ because from context confusion with multiplication seems unlikely. You can also consider putting $p$ and $b$ at a different corner, e.g. $E_n^{p,b}$ (the comma should make it obvious that it’s not an exponent) or $E_n^{(p,b)}$. These only work if you don’t want to take powers of the elements of your sequences. In the end, there probably is no perfect notation and you should just pick one. $\endgroup$ – Eike Schulte Dec 20 '18 at 5:41
  • $\begingroup$ What about simply refering to such a sequence as $\{pb^n\}_{n=1}^{\infty}$? This is easy to understand and does not introduce any complex notation. If I were reading a paper or book, I would greatly prefer the author to use $pb^n$ as the $n$th term, rather than $E(p,b)_n$ which seems designed only to make things harder to understand. Remember the KISS rule: en.wikipedia.org/wiki/KISS_principle $\endgroup$ – Michael Dec 20 '18 at 6:00
  • $\begingroup$ @Michael: Yes, I think that's the same as the last option I listed. My motivation for wanting to use a more-specific notation is that there are some transformations I want to perform on the sequence as a sequence, and I'll be drawing parallels to it from a different sequence that doesn't have a nice closed-form representation. So, in context, I don't think it would "seem[] designed only to make things harder to understand"; my only concern is with whether it would in fact be hard to understand. $\endgroup$ – ruakh Dec 20 '18 at 7:32
  • $\begingroup$ You could keep $(pb^n)$ in parentheses if you like, for the example context $$ T(\{a_n\}_{n=1}^{\infty}) = \sum_{n=1}^{\infty} a_nz^{-n} \implies T(\{pb^n\}_{n=1}^{\infty}) = \sum_{n=1}^{\infty} (pb^n)z^{-n}$$ and in fact "$(pb^n)$" uses fewer symbols than "$E(p,b)_n$." Now I have a similar question for you: $\endgroup$ – Michael Dec 20 '18 at 18:12
  • $\begingroup$ I am writing a paper with processes $Q_i(t)$ and $Z_i(t)$ defined for $t\in\{0, 1, 2,\ldots\}$ and $i \in \{1, \ldots, k\}$. Both have bounded changes from slot $t$ to slot $t+1$. I need notation for the maximum changes. Should it be: $$ \delta_{i,Q}^{max} = \sup_{t} |Q_i(t+1)-Q_i(t)|, \quad \delta_{i,Z}^{max} = \sup_t |Z_i(t+1)-Z_i(t)|$$ or $$ c_i^{max} = \sup_t |Q_i(t+1)-Q_i(t)|, \quad d_i^{max} = \sup_t|Z_i(t+1)-Z_i(t)|$$ or $$ q_i^{max} = \sup_t |Q_i(t+1)-Q_i(t)|, \quad z_i^{max} = \sup_t|Z_i(t+1)-Z_i(t)|$$ $\endgroup$ – Michael Dec 20 '18 at 18:13

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