Hölder's theorem

In mathematics, Hölder's theorem states that the gamma function does not satisfy any algebraic differential equation whose coefficients are rational functions. This result was first proved by Otto Hölder in 1887; several alternative proofs have subsequently been found.

The theorem also generalizes to the ${\displaystyle q}$-gamma function.

For every ${\displaystyle n\in \mathbb {N} _{0}}$, there is no non-zero polynomial ${\displaystyle P\in \mathbb {C} [X;Y_{0},Y_{1},\ldots ,Y_{n}]}$ such that

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