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  "body": "\n## Definition\n\nLet $K$ be a field of characteristic $0$, for example $\\C$.\nA [[formal power series|033WGN00BF]] $f \\in K[\\![x]\\!]$ is *holonomic* (or *D-finite*) if there are an integer $r \\ge 0$ and polynomials $p_0, p_1, \\ldots, p_r \\in K[x]$ with $p_r \\ne 0$ such that\n$$p_r(x) f^{(r)}(x) + \\cdots + p_1(x) f'(x) + p_0(x) f(x) = 0.$$\nHere $f^{(k)}$ is the $k$-th formal derivative of $f$ and $K[x]$ is the [[polynomial ring|0SDC934R52]].\nAn analytic function on an open set of $\\C$ is holonomic if it satisfies such an equation with $K = \\C$.\n\n## Remarks\n\nExample: $f(x) = \\frac{1}{1 - x} = \\sum_{n \\ge 0} x^n$ satisfies $(1 - x) f'(x) - f(x) = 0$, since $f'(x) = \\frac{1}{(1-x)^2}$; here $r = 1$, $p_1 = 1 - x$ and $p_0 = -1$.\nNon-example: $\\tan x$ is not holonomic.\nAn analytic holonomic function extends along paths avoiding the finitely many zeros of $p_r$, so it has only finitely many singularities, while $\\tan$ has poles at all points $\\frac{\\pi}{2} + k\\pi$.\nEquivalently, $f$ is holonomic if and only if the derivatives $f, f', f'', \\ldots$ span a finite-dimensional vector space over the field of [[rational functions|0XDK9CGYKJ]] $K(x)$.\nAnother equivalent form: $f = \\sum_n a_n x^n$ is holonomic if and only if its coefficients satisfy a [[recurrence|0FYX44GP92]] $q_d(n) a_{n+d} + \\cdots + q_0(n) a_n = 0$ for all $n$, with polynomials $q_i$ not all zero; such sequences are called P-recursive.\n\n## Explanations\n\n### L1 human\n\nA holonomic function is one that obeys a linear differential equation with polynomial coefficients.\nThe exponential $e^x$ is the simplest example: it satisfies $f' - f = 0$.\n",
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