How to perform upsampling (and NOT interpolation) process theoretically modelled?
If I understood your question right, you want a mathematical expression for the I/O (input/output) relationship of a signal expander (name of the block that expands (upsamples) -without interpolation filering- an input signal $x[n]$)
Below is a block diagram of signal expander by a factor of $L$:
$$ x[n] \longrightarrow \boxed{ \uparrow L } \longrightarrow x_e[n] \tag{1}$$
An expression for the expanded sequence $x_e[n]$ can be written as :
$$ x_e[n] = \begin{cases} { x[\frac{n}{L}] ~~~,~~~ n=m\cdot L ~~~,~~~ m=...,-1,0,1,... \\ ~~~ 0 ~~~~~~,~~ ~ \text{otherwise} } \end{cases} \tag{2} $$
An identical expression is also the following :
$$x_e[n] = \sum_{k=-\infty}^{\infty} x[k] \delta[n-L\cdot k] \tag{3}$$
where $\delta[n]$ is a unit sample (discrete-time impulse).
With $L=3$, an input $x[n]=[1,2,3,4]$ becomes output $x_e[n] =[1,0,0,2,0,0,3,0,0,4,0,0]$.