Interpolating and Decimating CIC filter based on sinc function
The frequency/z domain response of a $N$-Stage CIC filter is, invariably [1]:

$$\begin{align}
H(z) &amp;= \frac{(1-z^{-RM})^N}{(1-z^{-1})^N}\\
&amp;= \left(\sum\limits_{k=0}^{RM-1}{z^{-k}}\right)^N
\end{align}$$

with $R$ being the rate change, and $M$ being the delay length. You can assume $M=1$ (you don't get a special shape if you assume larger $M$, just sharper response).

For $N=1$, you get the "boring" observation that you've built a moving average filter (which, in fact, has sinc-shape in frequency domain). But that's really just due to the fact that that you really did just that – build a recursive MA.

The magnitude response is usually (as in [1]) represented as 

$$
\lvert H(f)\rvert = \left\lvert \frac{\sin(\pi M f)}{\pi M f} \right\rvert^N
$$

which goes to say you only get the sinc shape for $N=1$. In other words, a moving average. Which really shouldn't surprise you – the moving average is the time-domain rectangle, and the Fourier transform of that is the sinc.


[1] CIC Filter Introduction, Matthew P. Donadio, July 2000. Available Online