  A generalization of the Lyddane-Sachs-Teller relation is presented for polar
vibrations in materials with monoclinic and triclinic crystal systems. The
generalization is derived from an eigen displacement vector summation approach,
which is equivalent to the microscopic Born-Huang description of polar lattice
vibrations. An expression for a general oscillator strength is also described
for materials with monoclinic and triclinic crystal systems. A generalized
factorized dielectric response function characteristic for monoclinic and
triclinic materials is proposed. The generalized Lyddane-Sachs-Teller relation
is found valid for monoclinic $\beta$-Ga$_2$O$_3$, where accurate experimental
data became available recently from a comprehensive generalized ellipsometry
investigation. Data for triclinic crystal systems can be measured by
generalized ellipsometry as well, and are anticipated to become available soon
and results can be compared with the generalized relations presented here
