아페리(Apéry) 점화식

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
• $A_0=1, A_1=3, B_0=0, B_1=1$

$$ \begin{array}{cccc} \text{} & A_n & B_n & B_n/A_n \\ 0 & 1 & 0 & 0 \\ 1 & 3 & 1 & 0.3333333333 \\ 2 & 19 & \frac{25}{4} & 0.3289473684 \\ 3 & 147 & \frac{1741}{36} & 0.3289871504 \\ 4 & 1251 & \frac{6585}{16} & 0.3289868106 \\ 5 & 11253 & \frac{13327519}{3600} & 0.3289868134 \\ 6 & 104959 & \frac{124308457}{3600} & 0.3289868134 \\ 7 & 1004307 & \frac{19427741063}{58800} & 0.3289868134 \\ 8 & 9793891 & \frac{2273486234953}{705600} & 0.3289868134 \\ 9 & 96918753 & \frac{202482451324891}{6350400} & 0.3289868134 \\ \end{array} $$

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
• $\lim_{n\to \infty} \frac{B_n}{A_n} = \frac{1}{5}\zeta(2)$
• $|\frac{B_n}{A_n} -\frac{1}{5}\zeta(2)|= O(\lambda^{-2n})$ 여기서 $\lambda=(\frac{1+\sqrt{5}}{2})^5$

$\zeta(3)$

• 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
• $A_0=1, A_1=5, B_0=0, B_1=1$

$$ \begin{array}{cccc} \text{} & A_n & B_n & B_n/A_n \\ 0 & 1 & 0 & 0 \\ 1 & 5 & 1 & 0.2000000000 \\ 2 & 73 & \frac{117}{8} & 0.2003424658 \\ 3 & 1445 & \frac{62531}{216} & 0.2003428169 \\ 4 & 33001 & \frac{11424695}{1728} & 0.2003428172 \\ 5 & 819005 & \frac{35441662103}{216000} & 0.2003428172 \\ 6 & 21460825 & \frac{20637706271}{4800} & 0.2003428172 \\ 7 & 584307365 & \frac{963652602684713}{8232000} & 0.2003428172 \\ 8 & 16367912425 & \frac{43190915887542721}{13171200} & 0.2003428172 \\ 9 & 468690849005 & \frac{1502663969043851254939}{16003008000} & 0.2003428172 \\ \end{array} $$

• $A_n$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

• 점근 급수(asymptotic series)
• 홀로노믹 수열
• 무리수와 디오판투스 근사

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
  • http://jones.math.unibas.ch/~massierer/algebra-hs11/vanderPoorten2.pdf

관련논문

• 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.