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In this work, we analyze the three-body $B_{(s)} \to \eta_c(1S,2S) K \pi$decays within the framework of the perturbative QCD approach (PQCD) under thequasi-two-body approximation, where the kaon-pion invariant mass spectra aredominated by the$K_0^*(1430)^0,K_0^*(1950)^0,K^*(892)^0,K^*(1410)^0,K^*(1680)^0$ and$K_2^*(1430)^0$ resonances. The time-like form factors are adopted toparametrize the corresponding $S$, $P$, $D$-wave kaon-pion distributionamplitudes for the concerned decay modes, which describe the final-stateinteractions between the kaon and pion in the resonant region. The $K\pi$$S$-wave component at low $K\pi$ mass is described by the LASS line shape,while the time-like form factors of other resonances are modeled by therelativistic Breit-Wigner function. We find the following main points: (a) thePQCD predictions of the branching ratios for most considered $B \to\eta_c(1S)(K^{*0}\to )K^+\pi^-$ decays agree well with the currently availabledata within errors; (b) for ${\cal B}(B^0 \to \eta_c (K_0^*(1430)\to)K^+\pi^-)$ and ${\cal B}(B^0 \to \eta_c K^+\pi^-({\rm NR}))$ (here NR meansnonresonant), our predictions of the branching ratios are a bit smaller thanthe measured ones; and (c) the PQCD results for the $D$-wave contributionsconsidered in this work can be tested once the precise data from the futureLHCb and Belle-II experiments are available.

$S$, $P$ and $D$-wave resonance contributions to $B_{(s)} \to \eta_c(1S,2S) K\pi$ decays in the perturbative QCD approach