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.
[1] The theorem also generalizes to the
there is no non-zero polynomial
is the gamma function.
is called an algebraic differential equation, which, in this case, has the solutions
— the Bessel functions of the first and second kind respectively.
Most of the familiar special functions of mathematical physics are differentially algebraic.
All algebraic combinations of differentially algebraic functions are differentially algebraic.
Furthermore, all compositions of differentially algebraic functions are differentially algebraic.
Hölder’s Theorem simply states that the gamma function,
and assume that a non-zero polynomial
As a non-zero polynomial in
can never give rise to the zero function on any non-empty open domain of
(by the fundamental theorem of algebra), we may suppose, without loss of generality, that
contains a monomial term having a non-zero power of one of the indeterminates
has the lowest possible overall degree with respect to the lexicographic ordering
because the highest power of
in any monomial term of the first polynomial is smaller than that of the second polynomial.
If we define a second polynomial
then we obtain the following algebraic differential equation for
is the highest-degree monomial term in
, then the highest-degree monomial term in
has a smaller overall degree than
, and as it clearly gives rise to an algebraic differential equation for
, it must be the zero polynomial by the minimality assumption on
A change of variables then yields
and an application of mathematical induction (along with a change of variables at each induction step) to the earlier expression
, which contradicts the minimality assumption on
is not differentially algebraic.