Cyclotomic numbers and Chebyshev polynomials

introduction

• borrowed from Andrews-Gordon identity
• quantum dimension and thier recurrence relation

\[d_i=\frac{\sin \frac{(i+1)\pi}{k+2}}{\sin \frac{\pi}{k+2}}\] satisfies \[d_i^2=1+d_{i-1}d_{i+1}\] where \(d_0=1\), \(d_k=1\)

 

1. (*choose k for c (2,k+2) minimal model*)k := 11
   d[k_, i_] := Sin[(i + 1) Pi/(k + 2)]/Sin[Pi/(k + 2)]
   Table[{i, d[k, i]}, {i, 1, k}] // TableForm
   Table[{i, N[(d[k, i])^2 - (1 + d[k, i - 1]*d[k, i + 1]), 10]}, {i, 1,
      k}] // TableForm
   
2. Plot[d[k, i], {i, 0, 2 k}]
   

 

 

diagonals of polygon

Clear[r]
r[i_] := Sin[((i + 1) Pi)/7]/Sin[Pi/7]
Table[N[r[i], 10], {i, 0, 5}]
Table[N[r[i]^2 - (1 + r[i - 1] r[i + 1]), 10], {i, 1, 4}]

 

 

chebyshev polynomials

• 체비셰프 다항식
• http://mathworld.wolfram.com/ChebyshevPolynomialoftheSecondKind.html
  also obey the interesting determinant identity
  
  

 

 

history

• http://www.google.com/search?hl=en&tbs=tl:1&q=

 

 

related items

• sl(2) - orthogonal polynomials and Lie theory

 

 

articles

• Golden Fields: A Case for the Heptagon
  
  • Peter Steinbach, Mathematics Magazine Vol. 70, No. 1 (Feb., 1997), pp. 22-31