Hölder's theorem

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.[1] The theorem also generalizes to the {style d'affichage q} -gamma function.

Statement of the theorem For every {style d'affichage nin mathbb {N} _{0},} there is no non-zero polynomial {displaystyle Pin mathbb {C} [X;O_{0},O_{1},ldots ,O_{n}]} tel que {displaystyle forall zin mathbb {C} smallsetminus mathbb {Z} _{leq 0}:qquad Pleft(z;Gamma (z),Gamma '(z),ldots ,{Gamma ^{(n)}}(z)droit)=0,} où {style d'affichage Gamma } is the gamma function. {displaystyle quad blacksquare } Par exemple, définir {displaystyle Pin mathbb {C} [X;O_{0},O_{1},O_{2}]} par {displaystyle P~{empiler {texte{df}}{=}}~X^{2}O_{2}+XY_{1}+(X^{2}-nu ^{2})O_{0}.} Then the equation {displaystyle Pleft(z;F(z),F'(z),F''(z)droit)=z^{2}F''(z)+zf'(z)+la gauche(z ^{2}-nu ^{2}droit)F(z)équiv 0} is called an algebraic differential equation, qui, dans ce cas, has the solutions {displaystyle f=J_{nu }} et {displaystyle f=Y_{nu }} — the Bessel functions of the first and second kind respectively. Ainsi, nous disons que {displaystyle J_{nu }} et {style d'affichage Y_{nu }} are differentially algebraic (also algebraically transcendental). Most of the familiar special functions of mathematical physics are differentially algebraic. All algebraic combinations of differentially algebraic functions are differentially algebraic. Par ailleurs, all compositions of differentially algebraic functions are differentially algebraic. Hölder’s Theorem simply states that the gamma function, {style d'affichage Gamma } , is not differentially algebraic and is therefore transcendentally transcendental.[2] Proof Let {style d'affichage nin mathbb {N} _{0},} and assume that a non-zero polynomial {displaystyle Pin mathbb {C} [X;O_{0},O_{1},ldots ,O_{n}]} exists such that {displaystyle forall zin mathbb {C} smallsetminus mathbb {Z} _{leq 0}:qquad Pleft(z;Gamma (z),Gamma '(z),ldots ,{Gamma ^{(n)}}(z)droit)=0.} As a non-zero polynomial in {style d'affichage mathbb {C} [X]} can never give rise to the zero function on any non-empty open domain of {style d'affichage mathbb {C} } (by the Fundamental Theorem of Algebra), we may suppose, sans perte de généralité, ce {style d'affichage P} contains a monomial term having a non-zero power of one of the indeterminates {style d'affichage Y_{0},O_{1},ldots ,O_{n}} .

Assume also that {style d'affichage P} has the lowest possible overall degree with respect to the lexicographic ordering {style d'affichage Y_{0}displaystyle deg left(-3X^{10}O_{0}^{2}O_{1}^{4}+iX^{2}O_{2}droit)

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