  We study the low-energy behavior of the vertex function of a single Anderson
impurity away from half-filling for finite magnetic fields, using the Ward
identities with careful consideration of the anti-symmetry and analytic
properties. The asymptotic form of the vertex function
$\Gamma_{\sigma\sigma';\sigma'\sigma}^{}(i\omega,i\omega';i\omega',i\omega)$ is
determined up to terms of linear order with respect to the two frequencies
$\omega$ and $\omega'$, as well as the $\omega^2$ contribution for
anti-parallel spins $\sigma'\neq \sigma$ at $\omega'=0$. From these results, we
also obtain a series of the Fermi-liquid relations beyond those of
Yamada-Yosida. The $\omega^2$ real part of the self-energy
$\Sigma_{\sigma}^{}(i\omega)$ is shown to be expressed in terms of the double
derivative $\partial^2\Sigma_{\sigma}^{}(0)/\partial \epsilon_{d\sigma}^{2}$
with respect to the impurity energy level $\epsilon_{d\sigma}^{}$, and agrees
with the formula obtained recently by Filippone, Moca, von Delft, and Mora in
the Nozi\`{e}res phenomenological Fermi-liquid theory [Phys.\ Rev.\ B {\bf 95},
165404 (2017)]. We also calculate the $T^2$ correction of the self-energy, and
find that the real part can be expressed in terms of the three-body correlation
function $\chi_{\uparrow\downarrow,-\sigma}^{[3]} = \partial
\chi_{\uparrow\downarrow}/\partial \epsilon_{d,-\sigma}^{}$. We also provide an
alternative derivation of the asymptotic form of the vertex function.
Specifically, we calculate the skeleton diagrams for the vertex function
$\Gamma_{\sigma\sigma;\sigma\sigma}^{}(i\omega,0;0,i\omega)$ for parallel spins
up to order $U^4$ in the Coulomb repulsion $U$. It directly clarifies the fact
that the analytic components of order $\omega$ vanish as a result of the
cancellation of four related Feynman diagrams which are related to each other
through the anti-symmetry operation.
