Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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FACTORIZATION, IRREDUCIBILITY, AND SPLITTING PROPERTIES OF CHEBYSHEV POLYNOMIALS

  • Kyeonghee Jo (Division of Liberal Arts and Sciences Mokpo National Maritime University)
  • Received : 2024.11.28
  • Accepted : 2025.05.23
  • Published : 2025.10.31
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Abstract

This paper focuses on the properties and factorizations of the Chebyshev polynomials. We show that the Chebyshev polynomial sn(x) with the particular initial conditions s0(x) = 0, s1(x) = 1 has a simple factorization, and each factor in this factorization has a one-to-one correspondence with a factor of the integer n. More precisely, we define a family of polynomials Pk(x) with integer coefficients for each k = 1, 2, …, and show that they satisfy sn(x) = ∏d|n Pd(x) for all n = 1, 2, …. We also present some more results about the factorization properties and the irreducibility properties for the Chebyshev polynomials. Additionally, we establish a characterization for certain irreducible polynomials, including the Chebyshev polynomial vn(x) with the particular initial conditions v0(x) = 1, v1(x) = x - 1. We demonstrate that any nonconstant polynomial g(x) with integer coefficients that satisfies a specific identity, g(x)g(-x) = ±g(±(2-x2)), can be expressed as a product of irreducible polynomials Ψk(x), where Ψk is a polynomial such that ${\Psi}_k(x\,+\,{\frac{1}{x}})\,=\,x^{-{\frac{1}{2}}{\varphi}(k)}{\Phi}_k(x)$. (Here 𝚽k is the k-th cyclotomic polynomial and φ is Euler's totient function.)

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Acknowledgement

The author acknowledges support from the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (NRF-2021R1F1A1049444), the Open KIAS Center at the Korea Institute for Advanced Study, and the 2023 sabbatical year research grant from Mokpo National Maritime University. The author also thanks the referee for helpful comments and corrections.

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