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  "body": "\n## Definition\n\nLet $K$ be a field.\nA [[sequence|0G9EKM0E15]] $u \\colon \\N \\to K$ is *holonomic* (or *P-recursive*) if it satisfies some [[linear recurrence with polynomial coefficients|K6QV4WSTXX]].\nThat is, there are an integer $r \\ge 0$ and polynomials $p_0, \\dots, p_r \\in K[n]$, not all zero, with\n$$p_r(n)\\, u_{n+r} + \\cdots + p_0(n)\\, u_n = 0 \\quad \\text{for all } n \\in \\N.$$\n\n## Remarks\n\nExample: the factorials $u_n = n!$ form a holonomic sequence over $\\Q$, with $r = 1$, $p_1 = 1$ and $p_0 = -(n + 1)$, since $(n+1)! - (n+1) \\cdot n! = 0$.\nNon-example: $u_n = 2^{n^2}$ over $\\Q$ is not holonomic.\nIndeed, take a recurrence as above with $p_r \\ne 0$ and the other $p_i$ of degree at most $f$.\nIt gives $\\abs{u_{n+r}} \\le C n^{f} \\max_{i \u003c r} \\abs{u_{n+i}}$ for large $n$ and some constant $C$.\nSo $\\log \\abs{u_n}$ grows at most like a constant times $n \\log n$, while $\\log 2^{n^2} = n^2 \\log 2$.\nOver a field of characteristic $0$, $u$ is holonomic if and only if its generating function $\\sum u_n x^n$ is a [[holonomic function|7H44JD9NQD]].\nEvery solution of a linear recurrence with constant coefficients is holonomic.\n",
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