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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


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

© 1996-9 Eric W. Weisstein