D-finite power series

A formal power series $f(x)$ over a field of characteristic zero is D-finite if it satisfies a nontrivial linear differential equation with polynomial coefficients:The polynomials are not all zero. Equivalently, the coefficients of $f$ satisfy a linear recurrence relation with polynomial functions of the index as coefficients.

For example, $e^x$ satisfies $f^{\prime }-f=0$, and every rational function regular at zero is D-finite. The class is closed under sums, products, and derivatives. This permits symbolic computation with many generating functions using finite equations rather than an infinite list of coefficients.