함수 다이로그 항등식(functional dilogarithm identity)

수학노트
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개요

  • 로저스 다이로그 함수 (Rogers' dilogarithm)가 만족시키는 두 함수 항등식의 일반화

\[L(x)+L(1-xy)+L(y)+L\left(\frac{1-y}{1-xy}\right)+L\left(\frac{1-x}{1-xy} \right)=3L(1)\]

  • 클러스터 대수(cluster algebra) 를 이용하여 일반화됨
  • 가령 $A_n$ 딘킨 다이어그램으로부터, n 변수로 구성된 \((n^2+3n)/2\) 항 관계식을 찾을 수 있음
  • $2, 5, 9, 14, 20, 27, 35, 44, 54, 65,\cdots$


2항 관계식

  • \(S=\left\{x,\frac{1}{x}\right\}\)라 두면,

\[\sum_{a\in S}L(\frac{1}{1+a})=L\left(\frac{1}{\frac{1}{x}+1}\right)+L\left(\frac{1}{x+1}\right)=L(1)\]



5항 관계식

  • \(S=\left\{x,y,\frac{x+1}{y},\frac{y+1}{x},\frac{x+y+1}{x y}\right\}\) 이면,

\[\sum_{a\in S}L(\frac{1}{1+a})=L\left(\frac{1}{\frac{x+1}{y}+1}\right)+L\left(\frac{1}{\frac{y+1}{x}+1}\right)+L\left(\frac{1}{\frac{x+y+1}{x y}+1}\right)+L\left(\frac{1}{x+1}\right)+L\left(\frac{1}{y+1}\right)=2L(1)\]


9항 관계식

  • $S$를 다음과 같이 두자

\[ S=\left\{x,y,z,\frac{(x+1) (z+1)}{y},\frac{(x+y+1) (y+z+1)}{x y z},\\ \frac{x z+x+y+z+1}{x y},\frac{x z+x+y+z+1}{y z},\frac{y+1}{x},\frac{y+1}{z}\right\}\]

  • 다이로그 함수에 대하여 다음이 성립한다

\[\sum_{a\in S}L(\frac{1}{1+a})=3L(1)\]


14항 관계식

\[ \left\{x,z,\frac{(x+1) (z+1)}{y},\frac{z+1}{w},\frac{x z+x+y+z+1}{x y}, \frac{(w+z+1) (x z+x+y+z+1)}{w y z}\\,\frac{(y+z+1) (w (x+y+1)+x z+x+y+z+1)}{w x y z}, \frac{w (x+y+1)+x z+x+y+z+1}{y z},\\ \frac{w y+w+y+z+1}{w z},\frac{(x+y+1) (w y+w+y+z+1)}{x y z}, \frac{(w+1) (y+1)}{z},\frac{y+1}{x},w,y\right\} \] \[\sum_{a\in S}L(\frac{1}{1+a})=4L(1)\]


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관련논문

  • Tomoki Nakanishi, Rogers dilogarithms of higher degree and generalized cluster algebras, arXiv:1605.04777 [math.QA], May 16 2016, http://arxiv.org/abs/1605.04777
  • Soudères, Ismaël. “Functional Equations for Rogers Dilogarithm.” arXiv:1509.02869 [math], September 9, 2015. http://arxiv.org/abs/1509.02869.
  • Kerr, Matt, James D. Lewis, and Patrick Lopatto. “Simplicial Abel-Jacobi Maps and Reciprocity Laws.” arXiv:1502.05459 [math], February 18, 2015. http://arxiv.org/abs/1502.05459.
  • Herbert Gangl, Functional equations and ladders for polylogarithms http://www.maths.dur.ac.uk/~dma0hg/ladders2_ams.pdf
  • Nakanishi, Tomoki. 2011. “Dilogarithm Identities for Conformal Field Theories and Cluster Algebras: Simply Laced Case.” Nagoya Mathematical Journal 202 (June): 23–43. doi:10.1215/00277630-1260432.
  • Chapoton, Frédéric. 2005. “Functional Identities for the Rogers Dilogarithm Associated to Cluster Y-Systems.” Bulletin of the London Mathematical Society 37 (5) (October 1): 755 -760. doi:10.1112/S0024609305004510.
  • Herbert Gangl, Functional equations of polylogarithms http://www.mathematik.hu-berlin.de/~maphy/kyoto.pdf
  • Zagier, D. "Special Values and Functional Equations of Polylogarithms." Appendix A in Structural Properties of Polylogarithms (Ed. L. Lewin) http://people.mpim-bonn.mpg.de/zagier/files/tex/LewinPolylogarithms/fulltext.pdf
  • L.J. Rogers, On Function Sum Theorems Connected with the Series Formula Proc. London Math. Soc. (1907) s2-4(1): 169-189 doi:10.1112/plms/s2-4.1.169