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A P-recursive sequence satisfies a linear recurrence with polynomial coefficients:where the $p_j$ are polynomials and not all zero. The recurrence is required for all sufficiently large relevant integers, allowing exceptional initial indices. Constant-coefficient linear recurrences are a special case.

For $a_n=n!$, the recurrence $a_{n+1}-(n+1)a_n=0$ proves P-recursiveness. By contrast, a simple-looking nonlinear recurrence is not automatically P-recursive. Over a field of characteristic zero, the ordinary generating function of a P-recursive sequence is D-finite, linking discrete recurrences with linear differential equations having polynomial coefficients.