Digital Equations problem - The fundamental ones that everyone should know
D ear N atalie, as you also know that these are just letters of English (org Greek) alphabet which by themselves have no meaning or explanations other than what you have arbitrarily imposed on them and the following is by convention what's being imposed on them in the mathematics, physics and the standard DSP literature as accepted unless otherwise stated explicitly...


$\omega$ can represent both continuous-time and discrete-time frequencies in radians-per-second and radians-per-sample respectively. 
When continuous-time and discrete-time signals are used simultaneously (such as in sampling operations) then $\Omega$ represents continuous-time radians per second frequency and $\omega$ represents the discrete-time radians per sample frequency. 
$\Omega = 2 \pi f$ interprets $f$ as the continuous-time frequency in Hertz.
$\omega = 2 \pi f$ interprets $f$ the same (as frequency in Hertz) if $\omega$ was used for continuous-time radians per second frequency, otherwise if $\omega$ is used for discrete-time radians per sample frequency, then $f$ has no such interpretation.
If a discrete-time signal $x[n]$ is periodic, then its (fundamental) period is indicated by the capital letter $N$ indicating a minimum integer larger than $1$ for which $x[n] = x[n+N]$ holds.
if a continuous-time signal $x(t)$ is periodic, then its (fundamental) period is indicated by the capital letter $T_0$ indicating a minimum real number larger than $0$ for which $x(t) = x(t+T_0)$ holds. This $T_0$ can also be taken as $T$.
When a continuous-time signal is uniformly sampled by an (ideal) periodic impulse train, then the period of sampling is indicated by the real number $T_s$ seconds per sample.
The reciprocal of the sampling period $T_s$ is denoted as the sampling rate: $F_s = 1/T_s$ in samples per second. If $F_s$ considered as the sampling frequency, then $F_s$ is interpreted in Hz.
Letter $k$ is mostly used as the frequency integer index for the DFT or DFS $X[k]$ of a discrete-time sequence $x[n]$. It's also used to indicate the continuous-time Fourier series coefficient indexing. It's also used to indicate convolution sums index in discrete-time.