In part I, basic properties of a class of functions (called Psi-functions) are investigated, e.g., series representation, recurrence relation, asymptotic behavior. In part II, the solution of some versions of the diffusion equation by a series of Psi- and Omega-functions is treated. Psi-functions may be used as auxiliary functions in the case of one-dimensional non-stationary diffusion under removal process or forced convection.