Discrete Fourier Transform with different period

Hi all, I have a seemingly simple problem which is I'd like to efficiently evaluate the following sums:

$$Y_k = \sum_{j=0}^{n-1} c_j e^{\frac{i j k \alpha}{n}}$$

for $k=0...n-1$. Now if $\alpha = 2\pi$, then this reduces to a standard DFT and I can use a standard FFT library to compute the sums. But if $\alpha \ne 2\pi$ then I don't see how I can put this into standard DFT form to use a regular FFT library on this.

I guess this problem amounts to computing a DFT with harmonics that do not necessarily have periods of $2 \pi$.

Any help is appreciated!

AlephZero