We extend the k-inflation which is a type of kinetically driven inflationary model under the standard inflationary scenario to a possible warm inflationary scenario. The dynamical equations of this warm k-inflation model are obtained. We rewrite the slow-roll parameters which are different from the usual potential driven inflationary models and perform a linear stability analysis to give the proper slow-roll conditions in the warm k-inflation. Two cases, a power-law kinetic function and an exponential kinetic function, are studied, when the dissipative coefficient Γ = Γ 0 and Γ = Γ(φ), respectively. A proper number of e-folds is obtained in both concrete cases of warm k-inflation. We find a constant dissipative coefficient (Γ = Γ 0 ) is not a workable choice for these two cases while the two cases with Γ = Γ(φ) are self-consistent warm inflationary models. fered in standard inflation can be avoided. Furthermore, the slow-roll conditions are much more easily satisfied in warm inflationary scenario [12,13].

We investigate the cosmological perturbation theory and calculate the perturbation spectra in the warm kinflation. Unlike the scalar perturbations of k-inflation in the cold inflationary scenario, entropy perturbations should be considered in our model. We study how the entropy perturbations affect the evolution of curvature perturbation R on the comoving hypersurfaces and find the dissipation coefficient Γ plays a central role in the entropy perturbations and the evolution equation of R. In the weak dissipation condition, entropy perturbations on the large scale can be neglected such that the primordial spectrum of cosmological perturbations is due only to adiabatic perturbations. Like general warm inflationary models, density fluctuations also mainly originated from the thermal fluctuations of the inflaton field rather than the vacuum fluctuations in our model. Finally, we calculate the perturbation spectra and the tensor-to-scalar ratio, then find the tensor-to-scalar ratio is smaller than that in the k-inflation or the general potential-driven cold inflationary models.