The authors study the Cauchy problem for the space-time fractional diffusion equation
where is a sufficiently well-behaved function, is the Riesz-Feller space-fractional derivative of the order and the skewness , and is the Caputo time-fractional derivative of the order . If , then the condition (2) is supplemented by the additional condition
An analogon of the fundamental solution to the problem (1)–(2) (or (1)–(3)) is itroduced and determined via Fourier-Laplace transform:
A scaling property as well as the similarity relation are obtained for . It is found also the connection of the fundamental solution to the Mittag-Leffler function and to Mellin-Barnes integrals. Some particular cases are considered, namely space-fractional diffusion (), time-fractional diffusion (, ) and neutral diffusion (). A composition rule for is established in the case which ensures its probabilistic interpretation at its range. A general representation of the Green function in terms of Mellin-Barnes integrals is obtained. On its base explicit formulas for as well as asymptotics of the Green function for different values of the parameters are found. Qualitative remarks concerning the solvability of the space-fractional diffusion equation are made illustrated by plots describing the behaviour of the Green function and the fundamental solution to (1).