We theoretically investigate the dynamical stability of one-dimensional fullerene polymers by computing the phonon dispersion relations within the atomistic approach. We find that only seven models among 54 models proposed previously (Noda et al., 2015) are dynamically stable. We show that the low temperature specific heat of them is proportional to the square root of the temperature in a wider range of temperature compared to the case of single-walled carbon nanotubes. (C) 2018 Elsevier B.V. All rights reserved.