A255237 Array of conversion coefficients for the minimal polynomials C of 2 cos(Pi/n) in terms of Chebyshev's S-polynomials.

1, 2, 1, 0, 1, -1, 1, -1, 0, 1, 0, -1, 1, -2, 0, 1, 0, 0, -1, 1, 0, 0, -1, 0, 1, -1, -1, 0, 1, 2, 0, -2, 0, 1, 0, 0, 0, 0, -1, 1, -1, 0, -1, 0, 1, 0, 0, 0, 0, 0, -1, 1, -2, 0, 2, 0, -2, 0, 1, -1, -2, -1, 1, 1

Offset: 0

Author

Ahmet Zahid KÜÇÜK and Wolfdieter Lang, Mar 11 2015

Keywords

• sign
• easy
• tabf

Comments

The row length sequence is 1, 1 + A055034(n), n >= 1.

For the minimal polynomial C(n, x) of the algebraic number rho(n) := 2*cos(Pi/n) (the length ratio of the smallest diagonal and the side of a regular n-gon) see the coefficient array in A187360. The coefficient triangle of Chebyshev's S-polynomials is given in A049310.

The conversion is C(n, x) = sum(T(n, m)*S(m, x), m=0..delta(n)), for n >= 0 with C(0, x) := 1 (undefined product), delta(0) = 0 and delta(n) = A055034(n), n >= 1.

Originally Ahmet Zahid KÜÇÜK observed the structure for prime n. The precise formula for odd primes prime(n) = A000040(n), n >= 2, is C(prime(n), x) = S((prime(n)-1)/2, x) - S((prime(n)-3)/2, x).

This is equivalent to C(prime(n),x) = (-1)^((p(n)-1)/2)*S(prime(n)-1,I*sqrt(x-2)), with I^2 = -1.

Proof: The known identity S(n, x) - S(n-1, x) = (-1)^n*S(2*n, I*sqrt(x-2)) (from bisection). The degrees of the monic polynomials of both sides match, as do the known zeros.

The row sums give 1, 3, 1, 0, 0, 0, -1, 0, 0, -1, 1, 0, -1, 0, -1, -2, 0, 0, -1, 0 ...

The alternating row sums give 1, 1, -1, -2, 0, 2, -1, -2, 0, -1, 1, -2, -1, 2, -1, 0, 0, 2, -1, -2, ...

For the reverse problem, the factorization of S polynomials into C polynomials see a Apr 12 2018 comment in A049310. - Wolfdieter Lang, Apr 12 2018

Examples

			The array T(n, m) begins:
n\m   0  1  2  3  4  5  6 ...
0:    1
1:    2  1
2:    0  1
3:   -1  1
4:   -1  0  1
5:    0 -1  1
6:   -2  0  1
7:    0  0 -1  1
8:    0  0 -1  0  1
9:   -1 -1  0  1
10:   2  0 -2  0  1
11:   0  0  0  0 -1  1
12:  -1  0 -1  0  1
13:   0  0  0  0  0 -1  1
14:  -2  0  2  0 -2  0  1
15:  -1 -2 -1  1  1
...
n=0: C(0, x) = 1 = 1*S(0, x),
n=1: C(1, x) = 2 + x = 2*S(0, x) + 1*S(1, x),
n=2: C(2, x) = x = 0*S(0, x) + 1*S(1, x),
n=3: C(3, x) = -1 + x = -1*S(0, x) + 1*S(1, x),
n=4: C(4, x) = -2 + x^2 = -1*S(0, x) + 0 + 1*S(2, x) = -1 + (-1 + x^2), ...
		

Crossrefs

Cf. A187360, A055034, A049310.

Formula

The conversion is C(n, x) = sum(T(n, m)*S(m, x), m = 0..delta(n)), that is

T(n, m) = [S(m, x)] C(n, x), n >= 0, m = 0, ..., delta(n), with C(0, x) := 1, delta(0) = 0 and delta(n) = A055034(n), n >= 1. For the C and S polynomials see A187360 and A049310, respectively.

For n >= 2: T(prime(n), (prime(n) -1)/2) = +1, T(prime(n), (prime(n) -3)/2) = -1 and T(prime(n), m) = 0 otherwise.