What is the difference between y[n] and y(k) (using square brackets)?
The notation for discrete-time signals, $x[n]$, was first noticed by me in the original Oppenheim and Schaffer (1975), even though i didn't see that book until 1981.



So before, discrete-time signals in DSP used the same notation in mathematics used for sequences (infinite or not), with the subscript, $x_n$.  In both notations, the argument or subscript play the same role and are only integers.

$$ x[n] \triangleq x_n \qquad \qquad x\in\mathbb{C} \qquad n\in\mathbb{Z} $$

Usually the signal, $x[n]$, is real but it need not be.  But the argument, $n$, must be an integer which makes this no different than a sequence, $x_n$.

With the DTFT and DFT this notation was common pre-O&amp;S,

$$\begin{align}
 X_k &amp;\triangleq \sum\limits_{n=0}^{N-1} x_n e^{-i 2 \pi \frac{nk}{N}} \\
\\
 x_n &amp;= \frac{1}{N} \sum\limits_{k=0}^{N-1} X_k e^{+i 2 \pi \frac{nk}{N}} \\
\end{align}$$

or they would use that awful $W_N^{nk}$ notation.

Now the problem with the "$x_n$" notation is that you would get too many subscripts to deal with, if you had a signal that was a multi-element vector.  And there was nothing in the notation to denote which subscript was the "time-like" variable.  With analog signals, $v(t)$, this wasn't a problem.  If you had a cable with 8 wires (plus ground), you could label them "$v_1(t)$" to "$v_8(t)$" and looking at $v_m(t)$, there was no doubt that it meant the "$m$th element that is a function of time, $t$."  But with "$x_{m,n}$" (as opposed to the O&amp;S way "$x_m[n]$") there is nothing, other than order, differentiating discrete time from the "channel" number.  Add another dimension (such as the 2-dim pixel radiance as a function of time) and you have a worser mess:

$$ x_{r,c}[n] \qquad \text{vs.} \qquad x_{r,c,n} $$

$r$ is row number, $c$ is column number, and $n$ is discrete-time.

So with this notation, there is a parity between a vector of continuous-time (a.k.a. "analog") signals $x_m(t)$ and discrete-time (a.k.a. "digital") signals $x_m[n]$.  So, likely, whatever relationship one might see between the various analog signals $x_m(t)$ in a multi-conductor cable, I might expect to see between the various elements of the digital vector signal $x_m[n]$ and sampling would be no different than with a single-dimensional signal

$$ x_m[n] \triangleq x_m(t)\bigg|_{t=nT} \qquad \qquad T=\frac{1}{f_\mathrm{s}} $$

So this provides for a consistency of notation along with a way to differentiate arguments that must be integers whereas whatever is contained within parenths $f\big( \cdot \big)$ is considered to have no such restriction (could be a real or complex argument).  That is, to me, helpful.

So, if it's a very old DSP paper (pre-1978, perhaps), the notation you would see would almost certainly be $x_n$ for discrete-time signals and the DFT would be $X_k$ for discrete-frequency.  But this has evolved to 

$$\begin{align}
 X[k] &amp;\triangleq \sum\limits_{n=0}^{N-1} x[n] e^{-j 2 \pi \frac{nk}{N}} \\
\\
 x[n] &amp;= \frac{1}{N} \sum\limits_{k=0}^{N-1} X[k] e^{+j 2 \pi \frac{nk}{N}} \\
\end{align}$$

and EEs will use "$j$", instead of "$i$" for the imaginary unit.

Math folks might use this notation:

$$\begin{align}
 \hat{f}_k &amp;\triangleq \frac{1}{\sqrt{n}} \sum\limits_{j=0}^{n-1} f_j \, e^{-i 2 \pi \frac{jk}{n}} \\
\\
 f_j &amp;= \frac{1}{\sqrt{n}} \sum\limits_{k=0}^{n-1} \hat{f}_k \, e^{+i 2 \pi \frac{jk}{n}} \\
\end{align}$$