아페리(Apéry) 점화식

개요

• Ζ(3)는 무리수이다(아페리의 정리)의 증명에서 등장한 점화식

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)

관련 논문

• Don Zagier, Integral solutions of Apéry-like recurrence equations
• 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.
• 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