  We study the higher-order Fermi-liquid relations of Kondo systems for
arbitrary impurity-electron fillings, extending the many-body quantum
theoretical approach of Yamada-Yosida. It includes partly a microscopic
clarification of the related achievements based on Nozi\`{e}res'
phenomenological description: Filippone, Moca, von Delft, and Mora [Phys.\
Rev.\ B {\bf 95}, 165404 (2017)]. In our formulation, the Fermi-liquid
parameters such as the quasi-particle energy, damping, and transport
coefficients are related to each other through the total vertex
$\Gamma_{\sigma\sigma';\sigma'\sigma} (\omega, \omega'; \omega', \omega)$,
which may be regarded as a generalized Landau quasi-particle interaction. We
obtain exactly this function up to linear order with respect to the frequencies
$\omega$ and $\omega'$ using the anti-symmetry and analytic properties. The
coefficients acquire additional contributions of three-body fluctuations away
from half-filling through the non-linear susceptibilities. We also apply the
formulation to non-equilibrium transport through a quantum dot, and clarify how
the zero-bias peak evolves in a magnetic field.
