A181875 Numerator of coefficient array of minimal polynomials of cos(2Pi/n). Rising powers in x.

-1, 1, 1, 1, 1, 1, 0, 1, -1, 1, 1, -1, 1, -1, -1, 1, 1, -1, 0, 1, 1, -3, 0, 1, -1, -1, 1, 1, 3, -3, -1, 1, 1, -3, 0, 1, -1, 3, 3, -1, -5, 1, 1, 1, -1, -1, 1, 1, 1, -1, -1, 1, 1, 0, -1, 0, 1, 1, -1, -5, 5, 15, -3, -7, 1, 1, -1, -3, 0, 1, 1, 5, -5, -5, 15, 21, -7, -2, 1, 1, 5, 0, -5, 0, 1, 1, -1, 1, 3, -3, -1, 1, -1, 3, 3, -1, -1, 1, -1, -3, 15, 35, -35, -7, 7, 9, -9, -5, 1, 1, 1, 0, -1, 0, 1, -1, 5, 25, -5, -25, 1, 35, 0, -5, 0, 1, -1, -3, 3, 1, -5, -1, 1, 1, 9, 0, -15, 0, 27, 0, -9, 0, 1, -7, 0, 7, 0, -7, 0, 1, -1, 7, 7, -7, -63, 63, 105, -15, -165, 55, 33, -3, -13, 1, 1, 1, -1, -1, 1, 1

Offset: 1

Author

Wolfdieter Lang, Jan 08 2011

Keywords

• sign
• easy
• tabf

Comments

The corresponding denominator array is A181876(n,m).

The sequence of row lengths is d(n)+1, with d(n):=A023022(n), n>=2, and d(1):=1: [2, 2, 2, 2, 3, 2, 4, 3, 4, 3, 6, 3, 7, 4, 5, 5, 9, 4, 10, 5, 7,...].

Psi(n,x):=sum((a(n,m)/b(n,m))*x^m,m=0..d(n)), with the degree d(n):=A023022(n), n>=2, d(1):=1, and b(n,m):=A181876(n,m), is the minimal polynomial of cos(2*Pi/n), n>=1. For the definition of `minimal polynomial of an algebraic number' see, e.g., the I. Niven reference, p. 28 (monic, minimal degree rational polynomial with the algebraic number as one of its roots).

All the roots of the minimal polynomial Psi(n,x), are cos(2*Pi*k/n) for k from {0,1,...,floor(n/2)} and gcd(k,n)=1 (relatively prime). The degree d(n) (see above) of Psi(n,x), hence of the algebraic number cos(2*Pi/n), is 1 for n=1 and 2, and phi(n)/2 for n>2, with Euler's totient function phi(n)=A000010(n). See the D. H. Lehmer reference, and the I. Niven reference, Theorem 3.9, p. 37. This is the Lemma on p. 473 of the Watkins and Zeitlin reference (including the n=1 and n=2 cases).

A recurrence for Psi(n,x) is found in the Watkins and Zeitlin reference.

For the solution of the Watkins and Zeitlin recurrence see the W.Lang link under A007955, eqs. (1) and (3), and the theorem with proposition 1. W. Lang, Feb 26 2011.

The polynomials Psi(n,x), n=1..30, have been given in a comment on A023022 by A. Jasinski. See also the W. Lang link.

For powers of each prime number p one finds the following results for m=1,2,...:

1. p odd prime,p=2*k+1:(2^(k*p^(m-1)))*Psi(p^m,x) = 2*sum(T(l*p^(m-1),x),l=1..k) + 1, with Chebyshev's T-polynomials.

2. p=2, m=1: Psi(2,x) = x+1 = T(1,x) + 1.

For m=2,3,...:(2^(m-2))*Psi(2^m,x) = 2*T(2^(m-2),x).

For some odd p the case m=1 has been observed in an e-mail by G. Detlefs to W. Lang. Feb 26 2011.

For the proofs see the W. Lang link, note added.

D. Surowski and P. McCombs (see the reference) give in their theorem 3.1. an explicit formula for the (non-monic) minimal polynomial of 2*cos(2*Pi/p) for odd prime p, p=2*k+1, called Theta_p(x). Their formula checks with Theta_p(x)=(2^k)*Psi(p,x/2) (if the misprint sigma_{2k+1} is corrected to sigma_{2k-1}).

W. Lang, Feb 26 2011.

S. Beslin and V. de Angelis (see the reference) give an explicit formula for the (integer) minimal polynomial of sin(2*Pi/p), called S_p(x), and cos(2*Pi/p), called C_p(x), for odd prime p, p=2k+1, with the results:

S_p(x) = sum(((-1)^l)*binomial(p,2*l+1)*(1-x^2)^(k-l) *x^(2*l),l=0..k), and C_p(x) = S_p(sqrt((1-x)/2)).

C_p(x) checks with (2^k)*Psi(p,x) from the above formula for powers of p, with m=1. W. Lang, Feb 26 2011.

Examples

			Rows begin:
  [-1, 1],
  [1, 1],
  [1, 1],
  [0, 1],
  [-1, 1, 1],
  [-1, 1],
  [-1, -1, 1, 1],
  [-1, 0, 1],
  [1, -3, 0, 1],
  [-1, -1, 1],
  ...
Array of rationals a(n,m)/A181876(n,m):
  [-1, 1],
  [1, 1],
  [1/2, 1],
  [0, 1],
  [-1/4, 1/2, 1],
  [-1/2, 1],
  [-1/8, -1/2, 1/2, 1],
  [-1/2, 0, 1],
  [1/8, -3/4, 0, 1],
  [-1/4, -1/2, 1],
  ...
Psi(5,x) has the zeros cos(2*Pi/5)=(phi-1)/2 and cos(4*Pi/5)=-phi/2 with phi:=(1+sqrt(5))/2 (golden section).
		

References

• I. Niven, Irrational Numbers, The Math. Assoc. of America, second printing, 1963, distributed by John Wiley and Sons.

Links

• S. Beslin and V. de Angelis, The minimal Polynomials of sin(2pi/p) and cos(2pi/p), Mathematics Mag. 77.2 (2004) 146-9.
• Mamoru Doi, Polynomial sequences with the same recurrence relation as Chebyshev polynomials and the minimal polynomial of cos(2*pi/n), arXiv:2501.16478 [math.GM], 2025. See p. 12.
• Wolfdieter Lang, A181875/A181876. Minimal polynomials of cos(2Pi/n).
• Wolfdieter Lang, The field Q(2cos(pi/n)), its Galois group and length ratios in the regular n-gon, arXiv preprint arXiv:1210.1018 [math.GR], 2012-2017. - From _N. J. A. Sloane_, Dec 30 2012
• D. H. Lehmer, A Note on Trigonometric Algebraic Numbers, Am. Math. Monthly 40,3 (1933) 165-6.
• D. Surowski and P. McCombs, Homogeneous Polynomials and the Minimal Polynomials of cos(2pi/n), Missouri J. of Math. Sciences, 15,1 (2003) 4-14.
• William Watkins and Joel Zeitlin, The Minimal Polynomial of cos(2Pi/n), Am. Math. Monthly (1993) Vol. 100, No. 5, 471-4.

Crossrefs

Cf. A181876, A181877, A023022, A183918.

Programs

Mathematica

ro[n_] := Numerator[ cc = CoefficientList[ MinimalPolynomial[ Cos[2*Pi/n], x], x] ; cc / Last[cc]]; Flatten[ Table[ ro[n], {n, 1, 30}]] (* Jean-François Alcover, Sep 27 2011 *)

Formula

a(n,m) = numerator([x]^m Psi(n,x)), n>=1, m=0,1,..,d(n), with d(n):=A023022(n) and d(1):=1, where Psi(n,x) has been defined in the comment above and is given by Psi(n,x) = Product_{k=0..floor(n/2) and gcd(k,n)=1} (x-cos(2*Pi*k/n)), n>=1.