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 \(d_0=1\), \(d_k=1\)

 

diagonals of regular polygon

• length of hepagon

$$d_i = \frac{\sin (\pi (i+1)/7)}{\sin (\pi/7)} $$

 

chebyshev polynomials

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

 

 

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