  In this paper analysis of the concept of {\it associated homogeneous
distributions} (generalized functions) is given, and some problems related to
these distributions are solved. It is proved that (in the one-dimensional case)
there exist {\it only} {\it associated homogeneous distributions} of order
$k=1$. Next, we introduce a definition of {\it quasi associated homogeneous
distributions} and provide a mathematical description of all quasi associated
homogeneous distributions and their Fourier transform. It is proved that the
class of {\it quasi associated homogeneous distributions} coincides with the
class of distributions introduced by Gel$'$fand and Shilov \cite[Ch.I,\S
4.]{G-Sh} as the class of {\it associated homogeneous distributions}. For the
multidimensional case it is proved that $f$ is a {\it quasi associated
homogeneous distribution} if and only if it satisfies the Euler type system of
differential equations. A new type of $\Gamma$-functions generated by quasi
associated homogeneous distributions is defined.
