General elementary solution of a homogeneous $q$-sided convolution type equation

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