= P-recursive sequence
{c}
{title2=$\sum_{j=0}^r p_j(n)a_{n+j}=0$}
= P-recursive
{c}
{synonym}
= Polynomial-coefficient recurrence sequence
{synonym}
A <P-recursive sequence> satisfies a <linear recurrence> with <polynomial coefficients>:
$$
p_0(n)a_n+p_1(n)a_{n+1}+\cdots+p_r(n)a_{n+r}=0,
$$
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>.
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