  Functional Hamilton-Jacobi (HJ) equation, the central equation of the
holographic renormalization group (HRG), functional Schr\"{o}dinger equation,
and generalized Wilson-Polchinski (WP) equation, the central equation of the
functional renormalization group (FRG), are considered in $D$-dimensional
coordinate and abstract (formal) spaces. Instead of extra coordinates or an FRG
scale, a holographic scalar field $\varLambda$ is introduced. The extra
coordinate (or scale) is obtained as the amplitude of delta-field or constant
field configurations of $\varLambda$. A rigorous derivation of corresponding
integro-differential equation hierarchies for Green functions (GFs) as well as
the integration formula for functionals are given. Using the integration
formula, the functional (arbitrary configuration of $\varLambda$) solution for
the translation-invariant two-particle GF is obtained. For the delta-field and
the constant field configurations of $\varLambda$ this solution is studied in
detail. Separable solution for two-particle GF is briefly discussed. Then,
rigorous derivation of the quantum HJ and the continuity functional equations
from the functional Schr\"{o}dinger equation as well as the semiclassical
approximation are given. An iterative procedure for solving the functional
Schr\"{o}dinger equation is suggested. Translation-invariant solutions for
various GFs (both hierarchies) on delta-field configuration of $\varLambda$ are
obtained. In context of continuity equation and open quantum field systems an
optical potential is briefly discussed. Modes coarse graining growth functional
for WP functional is analyzed. An approximation scheme is proposed for the
generalized WP equation. With an optimized regulator translation-invariant
solutions for two-particle and four-particle amputated GFs from approximated WP
hierarchy are found analytically.
