홀로노믹 수열

개요

• 홀로노믹 수열 (P-recursive,P-finite 또는 D-finite이라고도 불림)
• 다음의 형태의 점화식

$$ c_k(n)a_{n+k}+c_{k-1}(n)a_{n+k-1}+\cdots+c_{0}(n)a_{n}=0 \label{lin} $$ 여기서 $c_0,\cdots, c_k\neq 0$는 $n$의 다항식

예

• 팩토리얼(factorial), $a_n=n!$

$$ a_{n+1}-(n+1)a_n=0 $$

• 카탈란 수열(Catalan numbers)

$$ (n+2)a_{n+1}+(-4 n-2)a_{n}=0 $$

• 아페리(Apéry) 점화식

$$ n^2 u_{n}-(11n^2-11n+3)u_{n-1}-(n-1)^2u_{n-2}=0 \label{z2} $$

매스매티카 파일 및 계산 리소스

• https://docs.google.com/file/d/0B8XXo8Tve1cxRjVVZU9kYTRMeFE/edit
• http://www.wolfram.com/products/mathematica/newin7/content/NewNumberTheoryCapabilities/WorkWithHolonomicSequences.html

리뷰, 에세이, 강의노트

• Koepf, Wolfram. 1997. “The Algebra of Holonomic Equations.” Mathematische Semesterberichte 44 (2): 173–194. doi:10.1007/s005910050032. http://www.mathematik.uni-kassel.de/~koepf/Publikationen/Algebra.pdf
• Some properties of holonomic sequences

관련논문

• 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.