Schaar's Identity

A generalization of the Gaussian Sum. For $p$ and $q$ of opposite Parity (i.e., one is Even and the other is Odd), Schaar's identity states

$${1\over\sqrt{q}}\sum_{r=0}^{q-1}e^{-\pi ir^2p/q} = {e^{-\pi i/4}\over\sqrt{p}}\sum_{r=0}^{p-1} e^{\pi ir^2q/p}.$$

See also Gaussian Sum

References

Evans, R. and Berndt, B. ``The Determination of Gauss Sums.'' Bull. Amer. Math. Soc. 5, 107-129, 1981.