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  "body": "\n## Definition\n\nLet $K$ be a field and let $r \\ge 0$ be an integer.\nLet $p_0, \\dots, p_r \\in K[n]$ be polynomials in the [[polynomial ring|0SDC934R52]] over $K$, with $p_r \\ne 0$.\nThe *linear recurrence with polynomial coefficients* $p_0, \\dots, p_r$ is the [[recurrence relation|07TZV87Y0T]]\n$$p_r(n)\\, u_{n+r} + p_{r-1}(n)\\, u_{n+r-1} + \\cdots + p_0(n)\\, u_n = 0 \\qquad (n \\in \\N)$$\nfor a sequence $u \\colon \\N \\to K$.\nIts order is $r$, when $p_0 \\ne 0$.\n\n## Remarks\n\nExample: the [[Catalan numbers|0YWZJEABB2]] $C_n$ satisfy $(n + 2)\\, C_{n+1} - (4n + 2)\\, C_n = 0$.\nCheck for $n = 0, 1$: $2 C_1 = 2 = 2 C_0$ and $3 C_2 = 6 = 6 C_1$, with $C_0 = C_1 = 1$ and $C_2 = 2$.\nNon-example: $u_{n+1} - 2^n u_n = 0$ fails the definition, since the coefficient $2^n$ is not a polynomial in $n$.\nWhen all $p_i$ are constant, the recurrence has constant coefficients.\nIf $p_r$ has a root $n_0 \\in \\N$, the equation for $n = n_0$ does not determine $u_{n_0 + r}$, so the initial values may not fix the solution.\nOther names are P-recursive or holonomic recurrence; an inhomogeneous version allows a polynomial right side.\n",
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