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Let \(\alpha _1, \ldots , \alpha _N\) and \(\beta _1, \ldots , \beta _N\) be two collections of numbers. If

then the monic polynomials with roots \(\{ \alpha _i\} \) and \(\{ \beta _i\} \) have identical coefficients of degree \(k \geq 1\). They may differ in the constant term if \(\sum \alpha _i \neq \sum \beta _i\) (i.e., if \(p_0\) differs).

When the \(N\)-th roots of unity are rotated by an angle \(\theta \) (equivalently, multiplied by \(\mathrm{e}^{i\theta }\)) and projected onto the real axis, we obtain the \(N\) real numbers

The rotation angle \(\theta \) controls the orientation of the \(N\)-gon, while the projection yields a multiset of real values in \([-1, 1]\).

To match the normalization of Chebyshev polynomials, which have leading coefficient \(2^{N-1}\) for \(N \geq 1\), we define the scaled polynomial:

The scaled polynomial inherits the degree and roots from \(P_N\).

Let \(P_N(x; \theta )\) denote the monic polynomial having the rotated roots as roots:

Since the roots are real and lie in \([-1, 1]\), this polynomial has real coefficients and degree \(N\), and each \(r_k(\theta )\) is a root.

For even \(m\) with \(0 {\lt} m {\lt} N\),

For odd \(m\) with \(0 {\lt} m {\lt} N\),

Let \(\xi _k = \cos ((2k + 1)\pi /(2N))\) for \(k = 0, \ldots , N-1\) denote the roots of \(T_N(x)\). Then for any \(1 \leq j {\lt} N\),

(See Lemma 6.1 in ../paper/chebyshev_circles.pdfthe paper)

The roots of the Chebyshev polynomial \(T_N(x)\) are

These are \(N\) distinct real numbers in the interval \((-1, 1)\).

For any integer \(m\),

(See ../docs/ChebyshevCircles/Proofs/ChebyshevOrthogonality.htmlAPI Documentation | ../paper/chebyshev_circles.pdfPaper Lemma 2.2)

Let \(r_k(\theta ) = \cos (\theta + 2\pi k/N)\) for \(k = 0, \ldots , N-1\). Then for any \(m\) with \(1 \leq m \leq N\), the elementary symmetric polynomial

is independent of \(\theta \).

For any integers \(N \geq 1\), \(j\) with \(1 \leq j {\lt} N\), and any \(\theta \in \mathbb {R}\),

is independent of \(\theta \).

(See ../docs/ChebyshevCircles/Proofs/PowerSums.html#powerSumCos_invariantAPI Documentation | ../paper/chebyshev_circles.pdfPaper Theorem 4.1)

For any integers \(N \geq 1\), \(m\) with \(0 {\lt} m {\lt} N\), and any \(\theta \in \mathbb {R}\),

(See Lemma 3.2 in ../paper/chebyshev_circles.pdfthe paper)

For any \(N \geq 2\) and any \(\theta \in \mathbb {R}\),

(See Lemma 3.1 in ../paper/chebyshev_circles.pdfthe paper)

The Chebyshev polynomials satisfy the following properties:

1. Recurrence: \(T_{n+1}(x) = 2xT_n(x) - T_{n-1}(x)\) for \(n \geq 1\).

2. Leading coefficient: The leading coefficient of \(T_N(x)\) is \(2^{N-1}\) for \(N \geq 1\).

3. Roots: The roots of \(T_N(x)\) are

   \begin{equation} \label{eq:chebyshev_roots} x_k = \cos \left(\frac{(2k + 1)\pi }{2N}\right), \quad k = 0, 1, \ldots , N-1. \end{equation}

   2

4. Extrema: On \([-1, 1]\), \(T_N\) attains its extremal values \(\pm 1\) at the \(N + 1\) points \(\cos (j\pi /N)\) for \(j = 0, 1, \ldots , N\).

For any \(N \geq 1\) and any \(k\) with \(1 \leq k \leq N\), the coefficient of \(x^k\) in \(S_N(x; \theta )\) is independent of \(\theta \).

(See Theorem 5.1 in ../paper/chebyshev_circles.pdfthe paper)

For any positive integer \(N \geq 1\) and any angle \(\theta \in \mathbb {R}\),

where the constant \(c(\theta )\) is given explicitly by

Moreover, all coefficients of \(S_N(x; \theta )\) of degree \(k \geq 1\) are independent of \(\theta \) and equal the corresponding coefficients of \(T_N(x)\).

The power sums and elementary symmetric polynomials are related by:

where we set \(e_0 = 1\).

(See ../docs/ChebyshevCircles/Proofs/NewtonIdentities.htmlAPI Documentation | ../paper/chebyshev_circles.pdfPaper Theorem 2.3)

For \(1 \leq j {\lt} N\),

(See Theorem 6.2 in ../paper/chebyshev_circles.pdfthe paper)