<p>We study initial value problems for a diffusion-wave equation with fractional order differentiation operator in time. The fractional differentiation operator is defined as a linear combination of two Dzhrbashyan–Nersesyan operators associated with two-parameter sequences. It is shown that the distribution of parameters in the sequences defining these operators influences the form and number of initial data and induces different initial value problems in general. In this paper, these problems are described, and explicit representations of their solutions are constructed.</p>