Discrete generating functions

The discrete generating function of one variable is defined as a generalization of discrete hypergeometric functions and some of its properties are investigated. This type of generating series uses a falling power in its definition as opposed to a monomial, and leads to solutions of delay difference equations with polynomial coefficients. In particular, the effect of the operator $\theta$, which is a modification of the forward difference operator $\Delta$, on the discrete generating functions is determined. Functional equations with the operator $\theta$ for difference generating functions of solutions to linear difference equations with constant and polynomial coefficients are derived. Finally, an analogue of differentiably finite ($D$-finite) power series is given for discrete power series and the condition for its $D$-finiteness is proven: the discrete generating function of $f(x)$ is $D$-finite if $f(x)$ is a polynomially recursive sequence (an analog of Stanley and Lipshits theorems).