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 -gamma function.
Statement of the theorem
For every there is no non-zero polynomial such that
For example, define by
Then the equation
Proof
Let and assume that a non-zero polynomial exists such that
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 .
Assume also that has the lowest possible overall degree with respect to the lexicographic ordering For example,
Next, observe that for all we have:
If we define a second polynomial by the transformation
Furthermore, if is the highest-degree monomial term in , then the highest-degree monomial term in is
Consequently, the polynomial
Now, let in to obtain
A change of variables then yields
This is possible only if is divisible by , which contradicts the minimality assumption on . Therefore, no such exists, and so is not differentially algebraic.[2][3] Q.E.D.
References
- ^ Bank, Steven B. & Kaufman, Robert. “A Note on Hölder’s Theorem Concerning the Gamma Function”, Mathematische Annalen, vol 232, 1978.
- ^ a b Rubel, Lee A. “A Survey of Transcendentally Transcendental Functions”, The American Mathematical Monthly 96: pp. 777–788 (November 1989). JSTOR 2324840
- ^ Boros, George & Moll, Victor. Irresistible Integrals, Cambridge University Press, 2004, Cambridge Books Online, 30 December 2011. doi:10.1017/CBO9780511617041.003