Factorization, irreducibility, and splitting properties of Chebyshev polynomials
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This paper focuses on the properties and factorizations of the Chebyshev polynomials. We show that the Chebyshev polynomial $s_n(x)$ with the particular initial conditions $s_0(x)=0, \,\, s_1(x)=1$ 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 $P_k(x)$ with integer coefficients for each $k=1,2,\ldots,$ and show that they satisfy $s_n(x)=\prod_{d\mid n}P_d(x)$ for all $n=1,2,\ldots$. 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 $v_n(x)$ with the particular initial conditions $v_0(x)=1, \,\, v_1(x)=x-1$. We demonstrate that any nonconstant polynomial $g(x)$ with integer coefficients that satisfies a specific identity, $g(x)g(-x)=\pm g(\pm(2-x^2))$, can be expressed as a product of irreducible polynomials $\Psi_k(x)$, where $\Psi_k$ is a polynomial such that $\Psi_k(x+\frac{1}{x})=x^{-\frac{1}{2}\varphi(k)}\Phi_k(x)$. (Here $\Phi_k$ is the $k$-th cyclotomic polynomial and $\varphi$ is Euler's totient function.)
Abstract
This paper focuses on the properties and factorizations of the Chebyshev polynomials. We show that the Chebyshev polynomial $s_n(x)$ with the particular initial conditions $s_0(x)=0, \,\, s_1(x)=1$ 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 $P_k(x)$ with integer coefficients for each $k=1,2,\ldots,$ and show that they satisfy $s_n(x)=\prod_{d\mid n}P_d(x)$ for all $n=1,2,\ldots$. 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 $v_n(x)$ with the particular initial conditions $v_0(x)=1, \,\, v_1(x)=x-1$. We demonstrate that any nonconstant polynomial $g(x)$ with integer coefficients that satisfies a specific identity, $g(x)g(-x)=\pm g(\pm(2-x^2))$, can be expressed as a product of irreducible polynomials $\Psi_k(x)$, where $\Psi_k$ is a polynomial such that $\Psi_k(x+\frac{1}{x})=x^{-\frac{1}{2}\varphi(k)}\Phi_k(x)$. (Here $\Phi_k$ is the $k$-th cyclotomic polynomial and $\varphi$ is Euler's totient function.)