"Cyclotomic numbers and Chebyshev polynomials"의 두 판 사이의 차이

2013년 8월 5일 (월) 04:31 판

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

(*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
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

@@ 2번째 줄: / 2번째 줄: @@
 * borrowed from [[Andrews-Gordon identity]]
-*  quantum dimension and thier recurrence relation<br><math>d_i=\frac{\sin \frac{(i+1)\pi}{k+2}}{\sin \frac{\pi}{k+2}}}</math> satisfies<br><math>d_i^2=1+d_{i-1}d_{i+1}</math> where <math>d_0=1</math>, <math>d_k=1</math><br>
+*  quantum dimension and thier recurrence relation
+:<math>d_i=\frac{\sin \frac{(i+1)\pi}{k+2}}{\sin \frac{\pi}{k+2}}</math> satisfies
+:<math>d_i^2=1+d_{i-1}d_{i+1}</math> where <math>d_0=1</math>, <math>d_k=1</math><br>
@@ 41번째 줄: / 43번째 줄: @@
 * [[sl(2) - orthogonal polynomials and Lie theory]]
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@@ 74번째 줄: / 53번째 줄: @@
 ** Peter Steinbach, Mathematics Magazine Vol. 70, No. 1 (Feb., 1997), pp. 22-31
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 [[분류:개인노트]]
 [[Category:research topics]]
 [[Category:quantum dimensions]]