Factoring Variants of Chebyshev Polynomials of the First and Second Kinds with Minimal Polynomials of cos(2π/d)

Factoring Variants of Chebyshev Polynomials with Minimal Polynomials of $\cos(\frac{2π}{d})$

“…We have solved the problem of factoring variants of Chebyshev polynomials of the third and fourth kinds, V n (x) ± 1 and W n (x) ± 1, in terms of minimal polynomials for cos( 2π d ). This was done by applying the method of Wolfram [12] for factoring T n (x) ± 1 and U n (x) ± 1 in a similar way. We have shown that there are no generalizations of this factorization to similar variants of Chebyshev polynomials of the fifth and sixth kinds, X n (x) ± 1 and Y n (x) ± 1.…”

“…In previous work, Wolfram [12] solved an open factorization problem for Chebyshev polynomials of the second kind U n (x) ± 1, and gave a more direct proof of the result for Chebyshev polynomials of the first kind, T n (x) ± 1 . We apply this method to solve the analogous factorization problems for Chebyshev polynomials of the third and fourth kinds.…”

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We solve the problem of factoring polynomials Vn(x)±1 and Wn(x)±1 where Vn(x) and Wn(x) are Chebyshev polynomials of the third and fourth kinds. The method of proof is based on previous work by Wolfram [12] for factoring variants of Chebyshev polynomials of the first and second kinds, Tn(x) ± 1 and Un(x) ± 1. We also show that, in general, there are no factorizations of variants of Chebyshev polynomials of the fifth and sixth kinds, Xn(x) ± 1 and Yn(x) ± 1 using minimal polynomials of cos( 2π d ).

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Irreducibility of Chebyshev-Lissajous polynomials

“…First, let us introduce a factorization of Chebyshev polynomials. For more factorizations, see [3]. Lemma 1.…”

Factoring Variants of Chebyshev Polynomials With Minimal Polynomials Of

“…In earlier work, Wolfram [17] solved an open factorisation problem for Chebyshev polynomials of the second kind U n (x) ± 1 and gave a more direct proof of the result for Chebyshev polynomials of the first kind, T n (x) ± 1. We apply this method to solve the analogous factorisation problems for Chebyshev polynomials of the third and fourth kinds.…”