First Differentatior and Integrator For discrete time series $Y$, first differentiation ($D_i=Y_i-Y_{i+1}$) and integrator ($S_i=Y_i+Y_{i+1}$) can be defined as two highpass and lowpass LTI digital filters. Where, transfer function for the first difference is:
$$
H(z)=1-z^{-1}
$$
and for the first integrator is:
$$
H(z)=1+z^{-1}
$$
These two transfer function corresponds to $\text{Conv}(Y,[1,-1])$ and $\text{Conv}(Y,[1,1])$ for differentiator and integrator, respectively.   

I was wondering what the power transfer function of these two filters should be.

I have to add, in terms of implementation in Matlab, I can get $S_i$ and $D_i$ of a white noise by:

Y = randn(1,100); 
D = diff(Y); 
S = Y(1:end-1)+Y(2:end);

By filter in Matlab:

S_Filt = filter([1  1],1,Y); 
D_Filt = filter([1 -1],1,Y);

Or by convolving conv the corresponding windows:

S_Conv = conv(Y,[1  1]);
D_Conv  = conv(Y,[1 -1]);

Note, all D* and S* variables should be identical (regardless of head and tail of time series, of course). 

Thanks in advance.
I think I found an answer to this question. So, I will post it here. 

For power transfer function of $1-z^{-1}$ we know:
\begin{eqnarray*}
1-z^{-1}&amp;=&amp;1-e^{-jw}\\
&amp;=&amp; 1-\cos(\omega)+j\sin(w)
\end{eqnarray*}
Then by using complex conjugate properties we have:
\begin{eqnarray*}
|1-z^{-1}|^2&amp;=&amp;1-2\cos(\omega)+\cos(\omega)^2+\sin(\omega)^2\\
&amp;=&amp;2(1-\cos(\omega)).
\end{eqnarray*}
And simillarly, for $1+z^{-1}$, we have:
\begin{eqnarray*}
|1+z^{-1}|^2&amp;=&amp; 1+2\cos(\omega)+\cos(\omega)^2+\sin(\omega)^2\\
&amp;=&amp; 2(1+\cos(\omega)).
\end{eqnarray*}

-

Note that the integrator in this question is a special case (see the expression for $S_i$) and is different than the conventional integrator $y[n]=y[n-1]+x[n]$.