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  "body": "\n## Statement\n\nLet $\\Gamma$ be the [[gamma function|0V3FZMVB4K]], as a holomorphic function on $\\C \\setminus \\set{0, -1, -2, \\ldots}$.\nLet $n \\ge 0$ be an integer and let $P \\in \\C[Y, X_0, X_1, \\ldots, X_n]$ be a polynomial in $n + 2$ variables (an element of a [[polynomial ring|0SDC934R52]]).\nIf\n$$P\\big(z, \\Gamma(z), \\Gamma'(z), \\ldots, \\Gamma^{(n)}(z)\\big) = 0$$\nfor all $z$ in the domain of $\\Gamma$, then $P = 0$.\n\n## Remarks\n\nIn words, $\\Gamma$ satisfies no algebraic differential equation whose coefficients are polynomials in $z$; one says that $\\Gamma$ is differentially transcendental.\nOne proof assumes a nonzero $P$ of least possible degree and uses the functional equation $\\Gamma(z + 1) = z\\Gamma(z)$ to build from it a nonzero polynomial relation of smaller degree, a contradiction.\nIn particular $\\Gamma$ is not a [[holonomic function|7H44JD9NQD]], since a linear differential equation with polynomial coefficients is a special algebraic one.\nBy contrast, the exponential and many other [[special functions|HWY72ZK273]], such as Bessel functions, do satisfy algebraic differential equations.\nThe theorem is named after Otto Hölder; it is not the same as [[Hölder's theorem on Archimedean groups|CDM40MT3HZ]].\n\n## Explanations\n\n### L1 human\n\nMany familiar functions obey a differential equation built from polynomials, like $f' = f$ for the exponential.\nHölder's theorem says the gamma function obeys no such equation, however complicated.\nSo it lives outside the world of functions described by polynomial differential equations.\n\n### L2 human\n\nThe theorem shows that the gamma function is not a solution of any of the classical differential equations of mathematical physics.\nThis contrasts with the hypergeometric and Bessel functions, which are defined by linear differential equations, and explains why $\\Gamma$ is usually studied through functional equations, integrals and products instead.\n",
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