A generalization of the Gaussian sum. For 
and 
of opposite parity (i.e., one is
even and the other is odd),
Schaar's identity states
 |
Schaar's identity can also be written so as to be valid for 
, 
with 
even.
Schaar's Identity
See also
Gaussian SumExplore with Wolfram|Alpha
References
Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987.Evans, R. and Berndt, B. "The Determination of Gauss Sums." Bull. Amer. Math. Soc. 5, 107-129, 1981.Referenced on Wolfram|Alpha
Schaar's IdentityCite this as:
Weisstein, Eric W. "Schaar's Identity." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SchaarsIdentity.html