  We study the spectral functions, the poles and their trajectories for
increasing $N_{c}$ of the vector kaon state $K^{\ast}(892),$ characterized by
$I(J^{P})=\frac{1}{2}(1^{-})$, and of the scalar kaons $K_{0}^{\ast}(800)$ and
$K_{0}^{\ast}(1430),$ characterized by $I(J^{P})=\frac{1}{2}(0^{+})$. To this
end, we use relativistic QFT's Lagrangians with both derivative and
non-derivative terms. In the vector kaonic sector the spectral function is well
approximated by a Breit-Wigner function: there is one single peak and,
correspondingly, a single pole in the complex plane. On the contrary, in the
scalar sector, although the Lagrangian contains only one scalar kaonic field,
we find two poles, one corresponding to a standard quark-antiquark
,,seed\textquotedblright\ state $K_{0}^{\ast}(1430),$ and one to a
\textquotedblleft companion\textquotedblright\ dynamically generate pole
$K_{0}^{\ast}(800)$. The latter does not correspond to any peak in the scalar
kaonic spectral function, but only to an enhancement in the low-energy regime.
