  Pion-kaon ($\pi K$) pairs occur frequently as final states in heavy-particle
decays. A consistent treatment of $\pi K$ scattering and production amplitudes
over a wide energy range is therefore mandatory for multiple applications: in
Standard Model tests; to describe crossed channels in the quest for exotic
hadronic states; and for an improved spectroscopy of excited kaon resonances.
In the elastic region, the phase shifts of $\pi K$ scattering in a given
partial wave are related to the phases of the respective $\pi K$ form factors
by Watson's theorem. Going beyond that, we here construct a representation of
the scalar $\pi K$ form factor that includes inelastic effects via resonance
exchange, while fulfilling all constraints from $\pi K$ scattering and
maintaining the correct analytic structure. As a first application, we consider
the decay ${\tau\to K_S\pi\nu_\tau}$, in particular, we study to which extent
the $S$-wave $K_0^*(1430)$ and the $P$-wave $K^*(1410)$ resonances can be
differentiated and provide an improved estimate of the $CP$ asymmetry produced
by a tensor operator. Finally, we extract the pole parameters of the
$K_0^*(1430)$ and $K_0^*(1950)$ resonances via Pad\'e approximants,
$\sqrt{s_{K_0^*(1430)}}=[1408(48)-i\, 180(48)]$ MeV and
$\sqrt{s_{K_0^*(1950)}}=[1863(12)-i\,136(20)]$ MeV, as well as the pole
residues. A generalization of the method also allows us to formally define a
branching fraction for ${\tau\to K_0^*(1430) \nu_\tau}$ in terms of the
corresponding residue, leading to the upper limit ${\text{BR}(\tau\to
K_0^*(1430) \nu_\tau)<1.6 \times 10^{-4}}$.
