Discrete Generating Functions

The notion of a discrete generating function is defined. The definition uses the falling factorial instead of a power function. A functional equation for the discrete generating function of a solution to a linear difference equation with constant coefficients is found. For the discrete generating function of a solution to a linear difference equation with polynomial coefficients, the notion of $\mathrm{D}$-finiteness is introduced and an analog of Stanley's theorem is proved; namely, a condition for the $\mathrm{D}$-finiteness of the discrete generating function of a solution to such an equation is obtained.