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A181875 Numerator of coefficient array of minimal polynomials of cos(2Pi/n). Rising powers in x. 13

%I #41 Oct 02 2023 13:48:24

%S -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,

%T -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,

%U -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

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

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

%C 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,...].

%C 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).

%C 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).

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

%C 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.

%C 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.

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

%C 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.

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

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

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

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

%C 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}).

%C W. Lang, Feb 26 2011.

%C 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:

%C 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 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.

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

%H S. Beslin and V. de Angelis, <a href="http://www.jstor.org/stable/3219105">The minimal Polynomials of sin(2pi/p) and cos(2pi/p)</a>, Mathematics Mag. 77.2 (2004) 146-9.

%H Wolfdieter Lang, <a href="/A181875/a181875_1.pdf">A181875/A181876. Minimal polynomials of cos(2Pi/n).</a>

%H Wolfdieter Lang, <a href="http://arxiv.org/abs/1210.1018">The field Q(2cos(pi/n)), its Galois group and length ratios in the regular n-gon</a>, arXiv preprint arXiv:1210.1018 [math.GR], 2012-2017. - From _N. J. A. Sloane_, Dec 30 2012

%H D. H. Lehmer, <a href="http://www.jstor.org/stable/2301023">A Note on Trigonometric Algebraic Numbers</a>, Am. Math. Monthly 40,3 (1933) 165-6.

%H D. Surowski and P. McCombs, <a href="http://www.math-cs.ucmo.edu/~mjms/2003.1/Surow.pdf">Homogeneous Polynomials and the Minimal Polynomials of cos(2pi/n)</a>, Missouri J. of Math. Sciences, 15,1 (2003) 4-14.

%H W. Watkins and J. Zeitlin, <a href="http://www.jstor.org/stable/2324301">The Minimal Polynomial of cos(2Pi/n)</a>, Am. Math. Monthly 100,5 (1993) 471-4.

%F 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(x-cos(2*Pi*k/n)),k=0..floor(n/2)and gcd(k,n)=1), n>=1.

%e Rows begin:

%e [-1, 1],

%e [1, 1],

%e [1, 1],

%e [0, 1],

%e [-1, 1, 1],

%e [-1, 1],

%e [-1, -1, 1, 1],

%e [-1, 0, 1],

%e [1, -3, 0, 1],

%e [-1, -1, 1],

%e ...

%e Array of rationals a(n,m)/A181876(n,m):

%e [-1, 1],

%e [1, 1],

%e [1/2, 1],

%e [0, 1],

%e [-1/4, 1/2, 1],

%e [-1/2, 1],

%e [-1/8, -1/2, 1/2, 1],

%e [-1/2, 0, 1],

%e [1/8, -3/4, 0, 1],

%e [-1/4, -1/2, 1],

%e ...

%e 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).

%t 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 *)

%Y Cf. A181876, A181877, A023022, A183918.

%K sign,easy,tabf

%O 1,22

%A _Wolfdieter Lang_, Jan 08 2011

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