Exponential polynomials satisfying a homogeneous equation of convolution type are called its elementary solutions. The article is devoted to convolution-type operators in the complex domain that generalize the well-known operators of
q
q
-sided convolution and
Ď€
\pi
-convolution. The properties of such operators are investigated and the general form of elementary solutions (general elementary solution) of a homogeneous equation of
q
q
-sided convolution type is described.