An explicit formula for the fourth moment of certain exponential sums

For integers m, n, q, k, with q,k≧1 and Dirichlet characters \(\chi, \chi' \text {\rm (mod}\,q)\) we define a generalized Kloosterman sum

$S(m,n,\chi, \chi', q)= \sideset{}{'} \sum_{a=1}^q \chi (a)G(a,\chi')e \left(\frac{ma^k+na}{q}\right)$

with a Dirichlet character and a Gauss sum G(a,χ′) as coefficient, where e(z)=e ^2πiz . The aim of this paper is to study the fourth power mean

$M_k(q)=\sum_m\ \sum_{\chi}\ \sum_{\chi'} \bigl|S(m,n,\chi,\chi', q)\bigr|^4$

obtaining explicit formulas for M _k (q).