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Yilda matematika, Xolder teoremasi deb ta'kidlaydi gamma funktsiyasi hech kimni qoniqtirmaydi algebraik differentsial tenglama ularning koeffitsientlari ratsional funktsiyalar. Ushbu natijani birinchi marta isbotladi Otto Xolder 1887 yilda; keyinchalik bir nechta muqobil dalillar topildi.^[1]

Teorema ham ni umumlashtiradi ${ displaystyle q}$-gamma funktsiyasi.

Teorema bayoni

Har bir kishi uchun ${ displaystyle n in mathbb {N} _ {0},}$ nolga teng bo'lmagan polinom yo'q ${ displaystyle P in mathbb {C} [X; Y_ {0}, Y_ {1}, ldots, Y_ {n}]}$ shu kabi

${ displaystyle forall z in mathbb {C} setminus mathbb {Z} _ { leq 0}: qquad P left (z; Gamma (z), Gamma '(z), ldots , { Gamma ^ {(n)}} (z) right) = 0,}$

qayerda ${ displaystyle Gamma}$ bo'ladi gamma funktsiyasi. ${ displaystyle quad blacksquare}$

Masalan, aniqlang ${ displaystyle P in mathbb {C} [X; Y_ {0}, Y_ {1}, Y_ {2}]}$ tomonidan

${ displaystyle P ~ { stackrel { text {df}} {=}} X ^ {2} Y_ {2} + XY_ {1} + (X ^ {2} - nu ^ {2}) Y_ { 0}.}$

Keyin tenglama

${ displaystyle P chap (z; f (z), f '(z), f' '(z) o'ng) = z ^ {2} f' '(z) + zf' (z) + chap (z ^ {2} - nu ^ {2} o'ng) f (z) equiv 0}$

deyiladi algebraik differentsial tenglama, bu holda, echimlarga ega ${ displaystyle f = J _ { nu}}$ va ${ displaystyle f = Y _ { nu}}$ - birinchi va ikkinchi turdagi Bessel funktsiyalari. Shuning uchun biz buni aytamiz ${ displaystyle J _ { nu}}$ va ${ displaystyle Y _ { nu}}$ bor differentsial algebraik (shuningdek algebraik transsendental). Matematik fizikaning ma'lum maxsus funktsiyalarining aksariyati differentsial algebraikdir. Differentsial algebraik funktsiyalarning barcha algebraik birikmalari differentsial algebraikdir. Bundan tashqari, differentsial algebraik funktsiyalarning barcha tarkibi differentsial algebraikdir. Xolder teoremasi shunchaki gamma funktsiyasi, ${ displaystyle Gamma}$, differentsial algebraik emas va shuning uchun ham transandantal ravishda transsendental.^[2]

Isbot

Ruxsat bering ${ displaystyle n in mathbb {N} _ {0},}$ va nolga teng bo'lmagan polinom deb taxmin qiling ${ displaystyle P in mathbb {C} [X; Y_ {0}, Y_ {1}, ldots, Y_ {n}]}$ shunday mavjud

${ displaystyle forall z in mathbb {C} setminus mathbb {Z} _ { leq 0}: qquad P left (z; Gamma (z), Gamma '(z), ldots , { Gamma ^ {(n)}} (z) o'ng) = 0.}$

In nolga teng bo'lmagan polinom sifatida ${ displaystyle mathbb {C} [X]}$ ning hech qanday bo'sh bo'lmagan ochiq domenida hech qachon nol funktsiyani keltirib chiqara olmaydi ${ displaystyle mathbb {C}}$ (Algebraning asosiy teoremasi bo'yicha), biz umumiylikni yo'qotmasdan, deb taxmin qilishimiz mumkin ${ displaystyle P}$ aniqlanmaganlardan birining nolga teng bo'lmagan kuchiga ega bo'lgan monomial atamani o'z ichiga oladi ${ displaystyle Y_ {0}, Y_ {1}, ldots, Y_ {n}}$.

Buni ham faraz qiling ${ displaystyle P}$ leksikografik buyurtma bo'yicha eng past darajadagi umumiy darajaga ega ${ displaystyle Y_ {0}$ Masalan,

${ displaystyle deg left (-3X ^ {10} Y_ {0} ^ {2} Y_ {1} ^ {4} + iX ^ {2} Y_ {2} right) < deg left (2XY_) {0} ^ {3} -Y_ {1} ^ {4} o'ng)}$

chunki eng yuqori kuch ${ displaystyle Y_ {0}}$ birinchi polinomning istalgan monomial muddatida ikkinchi polinomnikidan kichikroq.

Keyin, barchaga e'tibor bering ${ displaystyle z in mathbb {C} setminus mathbb {Z} _ { leq 0}}$ bizda ... bor:

${ displaystyle { begin {aligned} P chap (z + 1; Gamma (z + 1), Gamma '(z + 1), Gamma' '(z + 1), ldots, Gamma ^ {(n)} (z + 1) o'ng) & = P chap (z + 1; z Gamma (z), [z Gamma (z)] ', [z Gamma (z)]' ' , ldots, [z Gamma (z)] ^ {(n)} o'ng) & = P chap (z + 1; z Gamma (z), z Gamma '(z) + Gamma (z), z Gamma '' (z) +2 Gamma '(z), ldots, z { Gamma ^ {(n)}} (z) + n { Gamma ^ {(n-1) }} (z) right). end {hizalangan}}}$

Agar ikkinchi polinomni aniqlasak ${ displaystyle Q in mathbb {C} [X; Y_ {0}, Y_ {1}, ldots, Y_ {n}]}$ o'zgartirish orqali

${ displaystyle Q { stackrel { text {df}} {=}} ~ P (X + 1; XY_ {0}, XY_ {1} + Y_ {0}, XY_ {2} + 2Y_ {1}, ldots, XY_ {n} + nY_ {n-1}),}$

unda quyidagi algebraik differentsial tenglamani olamiz ${ displaystyle Gamma}$:

${ displaystyle forall z in mathbb {C} setminus mathbb {Z} _ { leq 0}: qquad Q chap (z; Gamma (z), Gamma '(z), ldots , { Gamma ^ {(n)}} (z) right) equiv 0.}$

Bundan tashqari, agar ${ displaystyle X ^ {h} Y_ {0} ^ {h_ {0}} Y_ {1} ^ {h_ {1}} cdots Y_ {n} ^ {h_ {n}}}$ ning eng yuqori darajadagi monomial atamasi ${ displaystyle P}$, keyin eng yuqori darajadagi monomial atama ${ displaystyle Q}$ bu

${ displaystyle X ^ {h + h_ {0} + h_ {1} + cdots + h_ {n}} Y_ {0} ^ {h_ {0}} Y_ {1} ^ {h_ {1}} cdots Y_ {n} ^ {h_ {n}}.}$

Binobarin, polinom

${ displaystyle Q-X ^ {h_ {0} + h_ {1} + cdots + h_ {n}} P}$

ga qaraganda kichikroq umumiy darajaga ega ${ displaystyle P}$va bu aniq uchun algebraik differentsial tenglamani keltirib chiqaradi ${ displaystyle Gamma}$, bu minimallik faraziga ko'ra nol polinom bo'lishi kerak ${ displaystyle P}$. Demak, aniqlovchi ${ displaystyle R in mathbb {C} [X]}$ tomonidan

${ displaystyle R { stackrel { text {df}} {=}} X ^ {h_ {0} + h_ {1} + cdots + h_ {n}},}$

biz olamiz

${ displaystyle Q = P (X + 1; XY_ {0}, XY_ {1} + Y_ {0}, XY_ {2} + 2Y_ {1}, ldots, XY_ {n} + nY_ {n-1} ) = R (X) cdot P (X; Y_ {0}, Y_ {1}, ldots, Y_ {n}).}$

Endi, ruxsat bering ${ displaystyle X = 0}$ yilda ${ displaystyle Q}$ olish

${ displaystyle Q (0; Y_ {0}, Y_ {1}, ldots, Y_ {n}) = P (1; 0, Y_ {0}, 2Y_ {1}, ldots, nY_ {n-1 }) = R (0) cdot P (0; Y_ {0}, Y_ {1}, ldots, Y_ {n}) = 0 _ { mathbb {C} [Y_ {0}, Y_ {1}, ldots, Y_ {n}]}.}$

Keyin o'zgaruvchilar o'zgarishi hosil beradi

${ displaystyle P (1; 0, Y_ {1}, Y_ {2}, ldots, Y_ {n}) = 0 _ { mathbb {C} [Y_ {0}, Y_ {1}, ldots, Y_ {n}]},}$

va oldingi matematikaga matematik induksiyani qo'llash (har bir induksiya bosqichida o'zgaruvchilar o'zgarishi bilan birga)

${ displaystyle P (X + 1; XY_ {0}, XY_ {1} + Y_ {0}, XY_ {2} + 2Y_ {1}, ldots, XY_ {n} + nY_ {n-1}) = R (X) cdot P (X; Y_ {0}, Y_ {1}, ldots, Y_ {n})}$

buni ochib beradi

${ displaystyle forall m in mathbb {N}: qquad P (m; 0, Y_ {1}, Y_ {2}, ldots, Y_ {n}) = 0 _ { mathbb {C} [Y_ {0}, Y_ {1}, ldots, Y_ {n}]}.}$

Bu faqat agar mumkin bo'lsa ${ displaystyle P}$ ga bo'linadi ${ displaystyle Y_ {0}}$, bu minimallik taxminiga zid keladi ${ displaystyle P}$. Shuning uchun, bunday emas ${ displaystyle P}$ mavjud va hokazo ${ displaystyle Gamma}$ differentsial algebraik emas.^[2][3] Q.E.D.

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