Gauss Sums forO(2n+1,q),

For a nontrivial additive characterλand a multiplicative characterχof the finite field withqelements (qis a power of an odd prime), the “Gauss” sum ∑λ(trw) overw SO(2n+1,q) and ∑χ(detw)λ(trw) overw O(2n+1,q) are considered. We show that both of them are constant multiples of the sum ∑λ(trw) overw Sp(2n, q), which is, according to our previous result, a polynomial inqwith coefficients involving powers of the usual Kloosterman sums. As a consequence, we can determine certain “signed generalized Kloosterman sums over nonsingular symmetric matrices,” which were previously determined by J. H. Hodges only in the case that one of the two arguments is zero.