$x^{(T)}(t)\overset{\Delta}{=}\left\{\begin{matrix} x(t) -\frac{T}{2}\leq t <\frac{T}{2}\\ 0 \quad \textrm{otherwise}\end{matrix}\right.$
Which relationship applies in general between the Fourier transforms $X^{(T)} (jω)$ and $X^{(nT)}(jω)$ of a T-periodic signal $x(t)$ for $n$ odd?
If $x(t)$ is periodic with period $T$, then $x^{(nT)}(t)$ simply consists of shifted copies of $x^{(T)}(t)$, which, for $n$ odd, can be written as the following convolution:
$$x^{(nT)}(t) = x^{(T)}(t) * \sum\limits_{k=-\frac{n-1}{2}}^{\frac{n-1}{2}} \delta (t-kT)$$
Taking the Fourier Transform of both sides yields:
$$\begin{align*} X^{(nT)}(j\omega) &= X^{(T)}(j\omega) \cdot \sum\limits_{k=-\frac{n-1}{2}}^{\frac{n-1}{2}} e^{-j\omega kT}\\ &= X^{(T)}(j\omega) \cdot \left[1+2\sum\limits_{k=1}^{\frac{n-1}{2}} \cos{(\omega kT)}\right]\\ \end{align*}$$