---
tag: SF3X43X837
kind: statement
role: theorem
name: Hölder's theorem
msc: [33B15, 12H05]
uses: [0V3FZMVB4K, 0SDC934R52]
about: [0H5SYBSED9]
sources:
  - url: https://en.wikipedia.org/wiki/H%C3%B6lder%27s_theorem
same_as:
  wikidata: Q1995468
  wikipedia: Hölder's theorem
---

## Statement

Let $\Gamma$ be the [[gamma function|0V3FZMVB4K]], as a holomorphic function on $\C \setminus \set{0, -1, -2, \ldots}$.
Let $n \ge 0$ 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]]).
If
$$P\big(z, \Gamma(z), \Gamma'(z), \ldots, \Gamma^{(n)}(z)\big) = 0$$
for all $z$ in the domain of $\Gamma$, then $P = 0$.

## Remarks

In words, $\Gamma$ satisfies no algebraic differential equation whose coefficients are polynomials in $z$; one says that $\Gamma$ is differentially transcendental.
One 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.
In particular $\Gamma$ is not a [[holonomic function|7H44JD9NQD]], since a linear differential equation with polynomial coefficients is a special algebraic one.
By contrast, the exponential and many other [[special functions|HWY72ZK273]], such as Bessel functions, do satisfy algebraic differential equations.
The theorem is named after Otto Hölder; it is not the same as [[Hölder's theorem on Archimedean groups|CDM40MT3HZ]].

## Explanations

### L1 human

Many familiar functions obey a differential equation built from polynomials, like $f' = f$ for the exponential.
Hölder's theorem says the gamma function obeys no such equation, however complicated.
So it lives outside the world of functions described by polynomial differential equations.

### L2 human

The theorem shows that the gamma function is not a solution of any of the classical differential equations of mathematical physics.
This 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.