"아페리(Apéry) 점화식"의 두 판 사이의 차이
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| (같은 사용자의 중간 판 8개는 보이지 않습니다) | |||
| 4번째 줄: | 4번째 줄: | ||
| − | == | + | ==<math>\zeta(2)</math>== |
* 다음 점화식을 생각하자 | * 다음 점화식을 생각하자 | ||
| − | + | :<math> | |
n^2 u_{n}-(11n^2-11n+3)u_{n-1}-(n-1)^2u_{n-2}=0 \label{z2} | n^2 u_{n}-(11n^2-11n+3)u_{n-1}-(n-1)^2u_{n-2}=0 \label{z2} | ||
| − | + | </math> | |
| − | * 수열 | + | * 수열 <math>A_n</math>과 <math>B_n</math>을 초기조건이 다음과 같이 주어진 점화식 \ref{z2}의 해라 하자 |
| − | * | + | * <math>A_0=1, A_1=3, B_0=0, B_1=5</math> |
| − | + | :<math> | |
\begin{array}{cccc} | \begin{array}{cccc} | ||
| − | + | n & A_n & B_n & B_n/A_n \\ | |
| + | \hline | ||
0 & 1 & 0 & 0 \\ | 0 & 1 & 0 & 0 \\ | ||
| − | 1 & 3 & 1 | + | 1 & 3 & 5 & 1.666666667 \\ |
| − | 2 & 19 & \frac{ | + | 2 & 19 & \frac{125}{4} & 1.644736842 \\ |
| − | 3 & 147 & \frac{ | + | 3 & 147 & \frac{8705}{36} & 1.644935752 \\ |
| − | 4 & 1251 & \frac{ | + | 4 & 1251 & \frac{32925}{16} & 1.644934053 \\ |
| − | 5 & 11253 & \frac{13327519}{ | + | 5 & 11253 & \frac{13327519}{720} & 1.644934067 \\ |
| − | 6 & 104959 & \frac{124308457}{ | + | 6 & 104959 & \frac{124308457}{720} & 1.644934067 \\ |
| − | 7 & 1004307 & \frac{19427741063}{ | + | 7 & 1004307 & \frac{19427741063}{11760} & 1.644934067 \\ |
| − | 8 & 9793891 & \frac{2273486234953}{ | + | 8 & 9793891 & \frac{2273486234953}{141120} & 1.644934067 \\ |
| − | 9 & 96918753 & \frac{202482451324891}{ | + | 9 & 96918753 & \frac{202482451324891}{1270080} & 1.644934067 \\ |
\end{array} | \end{array} | ||
| − | + | </math> | |
===아페리의 정리=== | ===아페리의 정리=== | ||
| − | * | + | * <math>A_n\in \mathbb{Z}</math> |
| − | * | + | * <math>D_n^2 B_n \in \mathbb{Z}</math> 여기서 <math>D_n\approx e^n</math>은 [[1부터 n까지의 최소공배수]] |
| − | * | + | * <math>\lim_{n\to \infty} \frac{B_n}{A_n} = \zeta(2)</math> |
| − | * | + | * <math>|\frac{B_n}{A_n} -\zeta(2)|= O(\lambda^{-2n})</math> 여기서 <math>\lambda=(\frac{1+\sqrt{5}}{2})^5\approx 11.0902</math> |
| + | ;따름정리 | ||
| + | <math>\zeta(2)</math>는 무리수이다 | ||
| + | ;증명 | ||
| + | 위의 정리로부터 다음을 얻는다 | ||
| + | :<math>D_n^2|B_n -A_n\zeta(2)|=O\left((\frac{e}{\lambda})^{2n}\right) \label{int1}</math> | ||
| + | ===증명의 아이디어=== | ||
| + | * 각 <math>n\in \mathbb{Z}_{\geq 0}</math>에 대하여 다음과 같이 유리함수 <math>R_n</math>을 정의하자 | ||
| + | :<math>R_n(t)=\frac{(-1)^n n! \prod_{j=1}^n (t - j)}{\prod_{j=0}^n (t + j)^2}</math> | ||
| + | * <math>t\to \infty</math>일 때, <math>R_n(t)=O(t^{-2})</math>이므로, <math>r_n=\sum_{m=1}^{\infty}R_n(m)</math>는 수렴한다 | ||
| + | * 이 때, 다음이 성립한다 | ||
| + | :<math> | ||
| + | r_n=A_n\zeta(2)-B(n) | ||
| + | </math> | ||
| + | ;증명 | ||
| + | 다음과 같이 <math>a_k, b_k\in \mathbb{Q}</math>를 정의하자 | ||
| + | :<math>a_k=\lim_{t\to -k}R_n(t)(t+k)^2</math> | ||
| + | :<math>b_k=\lim_{t\to -k}\frac{d}{dt}\left(R_n(t)(t+k)\right)</math> | ||
| + | 멱급수 <math>r_n(z)=\sum_{m=1}^{\infty}R_n(m)z^m</math>에 대하여 다음이 성립한다 | ||
| + | :<math> | ||
| + | r_{n}(z)=a(z)\sum_{l=1}^{\infty}\frac{z^l}{l^2}+b(z)(-\log(1-z))-c(z) | ||
| + | </math> | ||
| + | 여기서 | ||
| + | :<math> | ||
| + | a(z)=\sum_{k=0}^n a_k z^{-k}\\ | ||
| + | b(z)=\sum_{k=0}^n b_k z^{-k}\\ | ||
| + | c(z)=\sum_{k=0}^n a_k z^{-k}(\sum_{l=1}^k \frac{z^l}{l^2}+\sum_{k=0}^n b_k z^{-k}(\sum_{l=1}^k \frac{z^l}{l} | ||
| + | </math> | ||
| + | 다음이 성립한다 | ||
| + | :<math> | ||
| + | \lim_{z\to 1^{-}}r_n(z)=a(1)\zeta(2)-c(1), \, b(1)=0 | ||
| + | </math> | ||
| − | + | ==<math>\zeta(3)</math>== | |
| − | == | ||
* 다음 점화식을 생각하자 | * 다음 점화식을 생각하자 | ||
| − | + | :<math> | |
n^3u_n-(34 n^3 - 51 n^2 + 27 n - 5)u_{n-1}+(n-1)^3u_{n-2}=0\label{z3} | n^3u_n-(34 n^3 - 51 n^2 + 27 n - 5)u_{n-1}+(n-1)^3u_{n-2}=0\label{z3} | ||
| − | + | </math> | |
| − | * 수열 | + | * 수열 <math>A_n</math>과 <math>B_n</math>을 초기조건이 다음과 같이 주어진 점화식 \ref{z3}의 해라 하자 |
| − | * | + | * <math>A_0=1, A_1=5, B_0=0, B_1=1</math> |
| − | + | :<math> | |
\begin{array}{cccc} | \begin{array}{cccc} | ||
| − | + | n & A_n & B_n & B_n/A_n \\ | |
| + | \hline | ||
0 & 1 & 0 & 0 \\ | 0 & 1 & 0 & 0 \\ | ||
1 & 5 & 1 & 0.2000000000 \\ | 1 & 5 & 1 & 0.2000000000 \\ | ||
| 54번째 줄: | 86번째 줄: | ||
9 & 468690849005 & \frac{1502663969043851254939}{16003008000} & 0.2003428172 \\ | 9 & 468690849005 & \frac{1502663969043851254939}{16003008000} & 0.2003428172 \\ | ||
\end{array} | \end{array} | ||
| − | + | </math> | |
| − | * | + | * <math>A_n</math>의 점근 급수는 다음과 같이 주어진다 |
| − | + | :<math> | |
(17+12 \sqrt{2})^n n^{-3/2} \sum _{s=0}^{\infty } \frac{c_{1,s}}{n^s}+(17-12 \sqrt{2})^n n^{-3/2} \sum _{s=0}^{\infty } \frac{c_{2,s}}{n^s} | (17+12 \sqrt{2})^n n^{-3/2} \sum _{s=0}^{\infty } \frac{c_{1,s}}{n^s}+(17-12 \sqrt{2})^n n^{-3/2} \sum _{s=0}^{\infty } \frac{c_{2,s}}{n^s} | ||
| − | + | </math> | |
| − | 여기서 | + | 여기서 <math>c_{1,s},c_{2,s}</math>는 적당한 상수이며, |
| − | + | :<math>c_{1,0}=\frac{3+2 \sqrt{2}}{\pi ^{3/2} \left(4 \sqrt[4]{2}\right)}, \quad c_{2,0}=\frac{3-2 \sqrt{2}}{\pi ^{3/2} \left(4 \sqrt[4]{2}\right)}</math> | |
===아페리의 정리=== | ===아페리의 정리=== | ||
| − | * | + | * <math>A_n\in \mathbb{Z}</math> |
| − | * | + | * <math>D_n^3 B_n \in \mathbb{Z}</math> 여기서 <math>D_n\approx e^n</math>은 [[1부터 n까지의 최소공배수]] |
| − | * | + | * <math>\lim_{n\to \infty} \frac{B_n}{A_n} = \frac{1}{6}\zeta(3)</math> |
| − | * | + | * <math>|\frac{B_n}{A_n} -\frac{1}{6}\zeta(3)|= O(\lambda^{-2n})</math> 여기서 <math>\lambda=\left(1+\sqrt{2}\right)^4</math> |
| + | |||
==점화식== | ==점화식== | ||
| − | * | + | * <math>n^2 u_{n}-(An^2-An+\lambda)u_{n-1}+B(n-1)^2u_{n-2}=0</math> 꼴의 선형 점화식 |
| − | * | + | * <math>n^2 u_{n}-(7n^2-7n+2)u_{n-1}+8(n-1)^2u_{n-2}=0</math> |
| − | * | + | * <math>n^2 u_{n}-(11n^2-11n+3)u_{n-1}-(n-1)^2u_{n-2}=0</math> |
| 82번째 줄: | 115번째 줄: | ||
==에세이== | ==에세이== | ||
| − | * Frits Beukers [http://www.staff.science.uu.nl/~beuke106/caen.pdf Consequences of Apéry's work on | + | * Frits Beukers [http://www.staff.science.uu.nl/~beuke106/caen.pdf Consequences of Apéry's work on <math>\zeta(3)</math>], 2003 |
| − | * A. van der Poorten [http://dx.doi.org/10.1007/FBF03028234 A proof that Euler missed ... Apéry's Proof of the irrationality of | + | * A. van der Poorten [http://dx.doi.org/10.1007/FBF03028234 A proof that Euler missed ... Apéry's Proof of the irrationality of <math>\zeta(3)</math>], The Mathematical Intelligencer 1 (4): 195-203, 1979 |
| − | ** http:// | + | ** http://maths.mq.edu.au/~alf/45.pdf |
| − | |||
==관련논문== | ==관련논문== | ||
| + | * Ling Long, Robert Osburn, Holly Swisher, On a conjecture of Kimoto and Wakayama, http://arxiv.org/abs/1404.4723v2 | ||
| + | * Malik, Amita, and Armin Straub. “Divisibility Properties of Sporadic Ap’ery-like Numbers.” arXiv:1508.00297 [math], August 2, 2015. http://arxiv.org/abs/1508.00297. | ||
| + | * Moeller, Martin, and Don Zagier. ‘Modular Embeddings of Teichmueller Curves’. arXiv:1503.05690 [math], 19 March 2015. http://arxiv.org/abs/1503.05690. | ||
* Ekhad, Shalosh B., and Doron Zeilberger. “Searching for Apery-Style Miracles [Using, Inter-Alia, the Amazing Almkvist-Zeilberger Algorithm].” arXiv:1405.4445 [math], May 17, 2014. http://arxiv.org/abs/1405.4445. | * Ekhad, Shalosh B., and Doron Zeilberger. “Searching for Apery-Style Miracles [Using, Inter-Alia, the Amazing Almkvist-Zeilberger Algorithm].” arXiv:1405.4445 [math], May 17, 2014. http://arxiv.org/abs/1405.4445. | ||
* Golyshev, Vasily, and Masha Vlasenko. 2012. “Equations D3 and Spectral Elliptic Curves.” arXiv:1212.0205 [math] (December 2). http://arxiv.org/abs/1212.0205. | * Golyshev, Vasily, and Masha Vlasenko. 2012. “Equations D3 and Spectral Elliptic Curves.” arXiv:1212.0205 [math] (December 2). http://arxiv.org/abs/1212.0205. | ||
* Zagier, Don. 2009. “Integral Solutions of Apéry-like Recurrence Equations.” In Groups and Symmetries, 47:349–366. CRM Proc. Lecture Notes. Providence, RI: Amer. Math. Soc. http://people.mpim-bonn.mpg.de/zagier/files/tex/AperylikeRecEqs/fulltext.pdf | * Zagier, Don. 2009. “Integral Solutions of Apéry-like Recurrence Equations.” In Groups and Symmetries, 47:349–366. CRM Proc. Lecture Notes. Providence, RI: Amer. Math. Soc. http://people.mpim-bonn.mpg.de/zagier/files/tex/AperylikeRecEqs/fulltext.pdf | ||
| − | * Yang, Yifan. 2008. “Apéry Limits and Special Values of | + | * Yang, Yifan. 2008. “Apéry Limits and Special Values of <math>L</math>-Functions.” Journal of Mathematical Analysis and Applications 343 (1): 492–513. doi:10.1016/j.jmaa.2008.01.094. |
* Garoufalidis, Stavros. 2007. “An Ansatz for the Asymptotics of Hypergeometric Multisums.” arXiv:0706.0722 (June 5). http://arxiv.org/abs/0706.0722. | * Garoufalidis, Stavros. 2007. “An Ansatz for the Asymptotics of Hypergeometric Multisums.” arXiv:0706.0722 (June 5). http://arxiv.org/abs/0706.0722. | ||
* Beukers, Frits. 2002. “On Dwork’s Accessory Parameter Problem.” Mathematische Zeitschrift 241 (2): 425–444. doi:10.1007/s00209-002-0424-8. | * Beukers, Frits. 2002. “On Dwork’s Accessory Parameter Problem.” Mathematische Zeitschrift 241 (2): 425–444. doi:10.1007/s00209-002-0424-8. | ||
2020년 11월 12일 (목) 07:22 기준 최신판
개요
- Ζ(3)는 무리수이다(아페리의 정리)의 증명에서 등장한 선형점화식
- 무리수와 디오판투스 근사의 예
\(\zeta(2)\)
- 다음 점화식을 생각하자
\[ n^2 u_{n}-(11n^2-11n+3)u_{n-1}-(n-1)^2u_{n-2}=0 \label{z2} \]
- 수열 \(A_n\)과 \(B_n\)을 초기조건이 다음과 같이 주어진 점화식 \ref{z2}의 해라 하자
- \(A_0=1, A_1=3, B_0=0, B_1=5\)
\[ \begin{array}{cccc} n & A_n & B_n & B_n/A_n \\ \hline 0 & 1 & 0 & 0 \\ 1 & 3 & 5 & 1.666666667 \\ 2 & 19 & \frac{125}{4} & 1.644736842 \\ 3 & 147 & \frac{8705}{36} & 1.644935752 \\ 4 & 1251 & \frac{32925}{16} & 1.644934053 \\ 5 & 11253 & \frac{13327519}{720} & 1.644934067 \\ 6 & 104959 & \frac{124308457}{720} & 1.644934067 \\ 7 & 1004307 & \frac{19427741063}{11760} & 1.644934067 \\ 8 & 9793891 & \frac{2273486234953}{141120} & 1.644934067 \\ 9 & 96918753 & \frac{202482451324891}{1270080} & 1.644934067 \\ \end{array} \]
아페리의 정리
- \(A_n\in \mathbb{Z}\)
- \(D_n^2 B_n \in \mathbb{Z}\) 여기서 \(D_n\approx e^n\)은 1부터 n까지의 최소공배수
- \(\lim_{n\to \infty} \frac{B_n}{A_n} = \zeta(2)\)
- \(|\frac{B_n}{A_n} -\zeta(2)|= O(\lambda^{-2n})\) 여기서 \(\lambda=(\frac{1+\sqrt{5}}{2})^5\approx 11.0902\)
- 따름정리
\(\zeta(2)\)는 무리수이다
- 증명
위의 정리로부터 다음을 얻는다 \[D_n^2|B_n -A_n\zeta(2)|=O\left((\frac{e}{\lambda})^{2n}\right) \label{int1}\]
증명의 아이디어
- 각 \(n\in \mathbb{Z}_{\geq 0}\)에 대하여 다음과 같이 유리함수 \(R_n\)을 정의하자
\[R_n(t)=\frac{(-1)^n n! \prod_{j=1}^n (t - j)}{\prod_{j=0}^n (t + j)^2}\]
- \(t\to \infty\)일 때, \(R_n(t)=O(t^{-2})\)이므로, \(r_n=\sum_{m=1}^{\infty}R_n(m)\)는 수렴한다
- 이 때, 다음이 성립한다
\[ r_n=A_n\zeta(2)-B(n) \]
- 증명
다음과 같이 \(a_k, b_k\in \mathbb{Q}\)를 정의하자 \[a_k=\lim_{t\to -k}R_n(t)(t+k)^2\] \[b_k=\lim_{t\to -k}\frac{d}{dt}\left(R_n(t)(t+k)\right)\] 멱급수 \(r_n(z)=\sum_{m=1}^{\infty}R_n(m)z^m\)에 대하여 다음이 성립한다 \[ r_{n}(z)=a(z)\sum_{l=1}^{\infty}\frac{z^l}{l^2}+b(z)(-\log(1-z))-c(z) \] 여기서 \[ a(z)=\sum_{k=0}^n a_k z^{-k}\\ b(z)=\sum_{k=0}^n b_k z^{-k}\\ c(z)=\sum_{k=0}^n a_k z^{-k}(\sum_{l=1}^k \frac{z^l}{l^2}+\sum_{k=0}^n b_k z^{-k}(\sum_{l=1}^k \frac{z^l}{l} \] 다음이 성립한다 \[ \lim_{z\to 1^{-}}r_n(z)=a(1)\zeta(2)-c(1), \, b(1)=0 \]
\(\zeta(3)\)
- 다음 점화식을 생각하자
\[ n^3u_n-(34 n^3 - 51 n^2 + 27 n - 5)u_{n-1}+(n-1)^3u_{n-2}=0\label{z3} \]
- 수열 \(A_n\)과 \(B_n\)을 초기조건이 다음과 같이 주어진 점화식 \ref{z3}의 해라 하자
- \(A_0=1, A_1=5, B_0=0, B_1=1\)
\[ \begin{array}{cccc} n & A_n & B_n & B_n/A_n \\ \hline 0 & 1 & 0 & 0 \\ 1 & 5 & 1 & 0.2000000000 \\ 2 & 73 & \frac{117}{8} & 0.2003424658 \\ 3 & 1445 & \frac{62531}{216} & 0.2003428169 \\ 4 & 33001 & \frac{11424695}{1728} & 0.2003428172 \\ 5 & 819005 & \frac{35441662103}{216000} & 0.2003428172 \\ 6 & 21460825 & \frac{20637706271}{4800} & 0.2003428172 \\ 7 & 584307365 & \frac{963652602684713}{8232000} & 0.2003428172 \\ 8 & 16367912425 & \frac{43190915887542721}{13171200} & 0.2003428172 \\ 9 & 468690849005 & \frac{1502663969043851254939}{16003008000} & 0.2003428172 \\ \end{array} \]
- \(A_n\)의 점근 급수는 다음과 같이 주어진다
\[ (17+12 \sqrt{2})^n n^{-3/2} \sum _{s=0}^{\infty } \frac{c_{1,s}}{n^s}+(17-12 \sqrt{2})^n n^{-3/2} \sum _{s=0}^{\infty } \frac{c_{2,s}}{n^s} \] 여기서 \(c_{1,s},c_{2,s}\)는 적당한 상수이며, \[c_{1,0}=\frac{3+2 \sqrt{2}}{\pi ^{3/2} \left(4 \sqrt[4]{2}\right)}, \quad c_{2,0}=\frac{3-2 \sqrt{2}}{\pi ^{3/2} \left(4 \sqrt[4]{2}\right)}\]
아페리의 정리
- \(A_n\in \mathbb{Z}\)
- \(D_n^3 B_n \in \mathbb{Z}\) 여기서 \(D_n\approx e^n\)은 1부터 n까지의 최소공배수
- \(\lim_{n\to \infty} \frac{B_n}{A_n} = \frac{1}{6}\zeta(3)\)
- \(|\frac{B_n}{A_n} -\frac{1}{6}\zeta(3)|= O(\lambda^{-2n})\) 여기서 \(\lambda=\left(1+\sqrt{2}\right)^4\)
점화식
- \(n^2 u_{n}-(An^2-An+\lambda)u_{n-1}+B(n-1)^2u_{n-2}=0\) 꼴의 선형 점화식
- \(n^2 u_{n}-(7n^2-7n+2)u_{n-1}+8(n-1)^2u_{n-2}=0\)
- \(n^2 u_{n}-(11n^2-11n+3)u_{n-1}-(n-1)^2u_{n-2}=0\)
관련된 항목들
에세이
- Frits Beukers Consequences of Apéry's work on \(\zeta(3)\), 2003
- A. van der Poorten A proof that Euler missed ... Apéry's Proof of the irrationality of \(\zeta(3)\), The Mathematical Intelligencer 1 (4): 195-203, 1979
관련논문
- Ling Long, Robert Osburn, Holly Swisher, On a conjecture of Kimoto and Wakayama, http://arxiv.org/abs/1404.4723v2
- Malik, Amita, and Armin Straub. “Divisibility Properties of Sporadic Ap’ery-like Numbers.” arXiv:1508.00297 [math], August 2, 2015. http://arxiv.org/abs/1508.00297.
- Moeller, Martin, and Don Zagier. ‘Modular Embeddings of Teichmueller Curves’. arXiv:1503.05690 [math], 19 March 2015. http://arxiv.org/abs/1503.05690.
- Ekhad, Shalosh B., and Doron Zeilberger. “Searching for Apery-Style Miracles [Using, Inter-Alia, the Amazing Almkvist-Zeilberger Algorithm].” arXiv:1405.4445 [math], May 17, 2014. http://arxiv.org/abs/1405.4445.
- Golyshev, Vasily, and Masha Vlasenko. 2012. “Equations D3 and Spectral Elliptic Curves.” arXiv:1212.0205 [math] (December 2). http://arxiv.org/abs/1212.0205.
- Zagier, Don. 2009. “Integral Solutions of Apéry-like Recurrence Equations.” In Groups and Symmetries, 47:349–366. CRM Proc. Lecture Notes. Providence, RI: Amer. Math. Soc. http://people.mpim-bonn.mpg.de/zagier/files/tex/AperylikeRecEqs/fulltext.pdf
- Yang, Yifan. 2008. “Apéry Limits and Special Values of \(L\)-Functions.” Journal of Mathematical Analysis and Applications 343 (1): 492–513. doi:10.1016/j.jmaa.2008.01.094.
- Garoufalidis, Stavros. 2007. “An Ansatz for the Asymptotics of Hypergeometric Multisums.” arXiv:0706.0722 (June 5). http://arxiv.org/abs/0706.0722.
- Beukers, Frits. 2002. “On Dwork’s Accessory Parameter Problem.” Mathematische Zeitschrift 241 (2): 425–444. doi:10.1007/s00209-002-0424-8.
- McIntosh, Richard J. 1996. “An Asymptotic Formula for Binomial Sums.” Journal of Number Theory 58 (1) (May): 158–172. doi:10.1006/jnth.1996.0072.
- Wimp, Jet, and Doron Zeilberger. 1985. “Resurrecting the Asymptotics of Linear Recurrences.” Journal of Mathematical Analysis and Applications 111 (1) (October): 162–176. doi:10.1016/0022-247X(85)90209-4.