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 {displaystyle q} -gamma function.

Statement of the theorem For every {displaystyle nin mathbb {N} _{0},} there is no non-zero polynomial {displaystyle Pin mathbb {C} [X;Y_{0},Y_{1},ldots ,Y_{n}]} such that {displaystyle forall zin mathbb {C} smallsetminus mathbb {Z} _{leq 0}:qquad Pleft(z;Gamma (z),Gamma '(z),ldots ,{Gamma ^{(n)}}(z)right)=0,} where {displaystyle Gamma } is the gamma function. {displaystyle quad blacksquare } For example, define {displaystyle Pin mathbb {C} [X;Y_{0},Y_{1},Y_{2}]} by {displaystyle P~{stackrel {text{df}}{=}}~X^{2}Y_{2}+XY_{1}+(X^{2}-nu ^{2})Y_{0}.} Then the equation {displaystyle Pleft(z;f(z),f'(z),f''(z)right)=z^{2}f''(z)+zf'(z)+left(z^{2}-nu ^{2}right)f(z)equiv 0} is called an algebraic differential equation, which, in this case, has the solutions {displaystyle f=J_{nu }} and {displaystyle f=Y_{nu }} — the Bessel functions of the first and second kind respectively. Hence, we say that {displaystyle J_{nu }} and {displaystyle 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. Furthermore, all compositions of differentially algebraic functions are differentially algebraic. Hölder’s Theorem simply states that the gamma function, {displaystyle Gamma } , is not differentially algebraic and is therefore transcendentally transcendental.[2] Proof Let {displaystyle nin mathbb {N} _{0},} and assume that a non-zero polynomial {displaystyle Pin mathbb {C} [X;Y_{0},Y_{1},ldots ,Y_{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)right)=0.} As a non-zero polynomial in {displaystyle mathbb {C} [X]} can never give rise to the zero function on any non-empty open domain of {displaystyle mathbb {C} } (by the Fundamental Theorem of Algebra), we may suppose, without loss of generality, that {displaystyle P} contains a monomial term having a non-zero power of one of the indeterminates {displaystyle Y_{0},Y_{1},ldots ,Y_{n}} .

Assume also that {displaystyle P} has the lowest possible overall degree with respect to the lexicographic ordering {displaystyle Y_{0}

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