  The main motivation of our work is to create an efficient algorithm that
decides hypertranscendence of solutions of linear differential equations, via
the parameterized differential and Galois theories. To achieve this, we expand
the representation theory of linear differential algebraic groups and develop
new algorithms that calculate unipotent radicals of parameterized differential
Galois groups for differential equations whose coefficients are rational
functions. P. Berman and M.F. Singer presented an algorithm calculating the
differential Galois group for differential equations without parameters whose
differential operator is a composition of two completely reducible differential
operators. We use their algorithm as a part of our algorithm. As a result, we
find an effective criterion for the algebraic independence of the solutions of
parameterized differential equations and all of their derivatives with respect
to the parameter.
