How to write time waveform of transmitted sequence with certain chip rate?
While a reference to where you got this from would be helpful, I have a theory on where the term in question comes from. Note its form for increasing $n$:

$$
\exp\left({j\frac{\pi n}{2}}\right) = [1, j, -1, -j, 1, j, -1, -j, \ldots ], \ n = [0, 1, 2, 3, 4, 5, 6, 7, \ldots]
$$

This is a discrete-time complex sinusoid whose frequency is one-fourth of the sample rate. In digital communications signal processing, this is a very common intermediate frequency selection when using real-valued sampling. It is often chosen because the center of the desired passband lies in the center of the single independent sideband that you get when using real samples. Shifting the resulting signal to complex baseband is often a subsequent processing step.

In your case, the equation that you gave may express that you have a chip sequence that is modulated onto a complex-valued carrier at one-fourth of the sample rate. Since you don't indicate what the values of the signal are at time instants other than $nT_c$, it's hard to say whether this is actually the case.