Summation formulas for general Kloosterman sums

N. V. Kuznetsov's summation formula is generalized to the case of a discrete subgroup $G\subset SL_2(\mathbf R)$ and a system of multiplicators $\chi$, satisfying certain not too restrictive conditions. In the arithmetic cases, when $G$ is a congruence-subgroup in $SL_2(\mathbf Z)$, these conditions are satisfied. N. V. Kuznetsov's formula has been proved for the case $G=SL_2(\mathbf Z)$, $\chi=1$.