"P진 감마함수(p-adic gamma function)"의 두 판 사이의 차이
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==개요== | ==개요== | ||
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==정의== | ==정의== | ||
| − | + | * 자연수 <math>n</math> 에 대하여 다음과 같이 정의 | |
| − | 자연수 <math>n</math> 에 대하여 다음과 같이 정의 | + | :<math>\Gamma_p(n)=(-1)^n\prod_{(i,p)=1}^{n-1} i</math> |
| − | + | * 이를 <math>\mathbb{Z}_p</math>로 연속함수로 확장하여, p-adic 감마함수를 얻음 | |
| − | <math>\Gamma_p(n)=(-1)^n\prod_{(i,p)=1}^{n-1} i</math> | + | |
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| − | 이를 <math>\mathbb{Z}_p</math>로 연속함수로 확장하여, p-adic 감마함수를 얻음 | ||
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==기본적인 성질== | ==기본적인 성질== | ||
| + | * <math>x\in p\mathbb{Z}_p</math> 일 때, <math>\Gamma_p(x+1)=-\Gamma_p(x)</math> | ||
| + | * <math>x\not \in p\mathbb{Z}_p</math> 일 때, <math>\Gamma_p(x+1)=-x\Gamma_p(x)</math> | ||
| + | * <math>x \equiv y \pmod {p^r}</math> 이면 <math>\Gamma_p(x)\equiv \Gamma_p(y) \pmod {p^r}</math> | ||
| + | * <math>p>3</math> 이면 <math>|\Gamma_p(x)-\Gamma_p(y)|_p \leq |x-y|_p</math> | ||
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==반사공식== | ==반사공식== | ||
| − | + | * <math>p\neq 2</math>이고, <math>x\in \mathbb{Z}_p</math> 에 대하여 다음 반사공식이 성립 | |
| − | <math>p\neq 2</math>이고, <math>x\in \mathbb{Z}_p</math> 에 대하여 다음 반사공식이 성립 | + | :<math>\Gamma_p(x)\Gamma_p(1-x)=(-1)^{l(x)}</math> |
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| − | <math>\Gamma_p(x)\Gamma_p(1-x)=(-1)^{l(x)}</math> | ||
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여기서 <math>x\equiv l(x) \pmod p</math>, <math>l(x)\in \{1,2,\cdots, p\}</math> | 여기서 <math>x\equiv l(x) \pmod p</math>, <math>l(x)\in \{1,2,\cdots, p\}</math> | ||
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==역사== | ==역사== | ||
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* [[수학사 연표]] | * [[수학사 연표]] | ||
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==관련된 항목들== | ==관련된 항목들== | ||
| − | + | * [[P진 해석학(p-adic analysis)]] | |
* [[감마함수]] | * [[감마함수]] | ||
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==관련논문== | ==관련논문== | ||
| − | * [http://www.numdam.org/item?id=RSMUP_2001__105__157_0 The Gross Koblitz formula revisited] | + | * [http://www.numdam.org/item?id=RSMUP_2001__105__157_0 The Gross Koblitz formula revisited] |
** Robert, Alain M. , Rend. Sem. Mat. Univ. Padova 105 (2001) 157 170. | ** Robert, Alain M. , Rend. Sem. Mat. Univ. Padova 105 (2001) 157 170. | ||
| − | * [http://www.springerlink.com/content/bq28602x02m17760/ p-adic gamma functions and their applications] | + | * [http://www.springerlink.com/content/bq28602x02m17760/ p-adic gamma functions and their applications] |
** Jack Diamond, 1984 | ** Jack Diamond, 1984 | ||
| − | * [http://archive.numdam.org/ARCHIVE/GAU/GAU_1981-1982__9_3/GAU_1981-1982__9_3_A18_0/GAU_1981-1982__9_3_A18_0.pdf The p-adic gamma function and congruences of Atkin and. Swinnerton-Dyer] | + | * [http://archive.numdam.org/ARCHIVE/GAU/GAU_1981-1982__9_3/GAU_1981-1982__9_3_A18_0/GAU_1981-1982__9_3_A18_0.pdf The p-adic gamma function and congruences of Atkin and. Swinnerton-Dyer] |
** L. van Hamme, Groupe d'étude d'analyse ultramétrique, 9e année 81/82, Fasc. 3 no. J17-6p | ** L. van Hamme, Groupe d'étude d'analyse ultramétrique, 9e année 81/82, Fasc. 3 no. J17-6p | ||
| − | * [http://www.jstor.org/stable/1971226 Gauss Sums and the p-adic Γ-function] | + | * [http://www.jstor.org/stable/1971226 Gauss Sums and the p-adic Γ-function] |
** Benedict H. Gross and Neal Koblitz, The Annals of Mathematics, Second Series, Vol. 109, No. 3 (May, 1979), pp. 569-581 | ** Benedict H. Gross and Neal Koblitz, The Annals of Mathematics, Second Series, Vol. 109, No. 3 (May, 1979), pp. 569-581 | ||
| − | * [http://www.jstor.org/stable/1997840 The $p$-Adic Log Gamma Function and $p$-Adic Euler Constants] | + | * [http://www.jstor.org/stable/1997840 The $p$-Adic Log Gamma Function and $p$-Adic Euler Constants] |
| − | ** Jack Diamond, | + | ** Jack Diamond, Transactions of the American Mathematical Society, Vol. 233, (Oct., 1977), pp. 321-337 |
| − | * [http://hdl.handle.net/2261/6494 A p-adic analogue of the $\Gamma$-function] | + | * [http://hdl.handle.net/2261/6494 A p-adic analogue of the $\Gamma$-function] |
** Morita, Yasuo, Journal of the Faculty of Science, the University of Tokyo. Sect. 1 A, Mathematics, Vol.22(1975), No.2, Page 255-266 | ** Morita, Yasuo, Journal of the Faculty of Science, the University of Tokyo. Sect. 1 A, Mathematics, Vol.22(1975), No.2, Page 255-266 | ||
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| − | + | [[분류:정수론]] | |
2014년 7월 12일 (토) 07:32 판
개요
정의
- 자연수 \(n\) 에 대하여 다음과 같이 정의
\[\Gamma_p(n)=(-1)^n\prod_{(i,p)=1}^{n-1} i\]
- 이를 \(\mathbb{Z}_p\)로 연속함수로 확장하여, p-adic 감마함수를 얻음
기본적인 성질
- \(x\in p\mathbb{Z}_p\) 일 때, \(\Gamma_p(x+1)=-\Gamma_p(x)\)
- \(x\not \in p\mathbb{Z}_p\) 일 때, \(\Gamma_p(x+1)=-x\Gamma_p(x)\)
- \(x \equiv y \pmod {p^r}\) 이면 \(\Gamma_p(x)\equiv \Gamma_p(y) \pmod {p^r}\)
- \(p>3\) 이면 \(|\Gamma_p(x)-\Gamma_p(y)|_p \leq |x-y|_p\)
반사공식
- \(p\neq 2\)이고, \(x\in \mathbb{Z}_p\) 에 대하여 다음 반사공식이 성립
\[\Gamma_p(x)\Gamma_p(1-x)=(-1)^{l(x)}\] 여기서 \(x\equiv l(x) \pmod p\), \(l(x)\in \{1,2,\cdots, p\}\)
역사
관련된 항목들
관련논문
- The Gross Koblitz formula revisited
- Robert, Alain M. , Rend. Sem. Mat. Univ. Padova 105 (2001) 157 170.
- p-adic gamma functions and their applications
- Jack Diamond, 1984
- The p-adic gamma function and congruences of Atkin and. Swinnerton-Dyer
- L. van Hamme, Groupe d'étude d'analyse ultramétrique, 9e année 81/82, Fasc. 3 no. J17-6p
- Gauss Sums and the p-adic Γ-function
- Benedict H. Gross and Neal Koblitz, The Annals of Mathematics, Second Series, Vol. 109, No. 3 (May, 1979), pp. 569-581
- The $p$-Adic Log Gamma Function and $p$-Adic Euler Constants
- Jack Diamond, Transactions of the American Mathematical Society, Vol. 233, (Oct., 1977), pp. 321-337
- A p-adic analogue of the $\Gamma$-function
- Morita, Yasuo, Journal of the Faculty of Science, the University of Tokyo. Sect. 1 A, Mathematics, Vol.22(1975), No.2, Page 255-266