Some properties of the quadrinomials $p(z)=1+κ(z+z^{N-1})+z^N$ and $q(z)=1+κ(z-z^{N-1})-z^N$
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| Formato: | Preprint |
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2024
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| author | Dmitrishin, Dmitriy Stokolos, Alexander |
| author_facet | Dmitrishin, Dmitriy Stokolos, Alexander |
| contents | We show that all the zeros of the quadrinomial $p(z)=1+κ(z+z^{N-1})+z^N$ lie on the unit circle if and only if the inequalities \[ -1\leκ\le 1\; (\mbox{ if $N$ is even}),\;\; -1\leκ\le N/(N-2)\; (\mbox{ if $N$ is odd}) \] hold. For the quadrinomial $q(z)=1+κ(z-z^{N-1})-z^N$, the corresponding inequalities are \[
-N/(N-2)\leκ\le 1\; (\text{ if $N$ is odd}),\;\;
-N/(N-2)\leκ\le N/(N-2)\; (\text{ if $N$ is even}). \] In the cases of limiting values of the parameter $κ$, we provide factorization formulas for the corresponding quadrinomials. For example, when $N$ is odd and $κ=N/(N-2)$, the following representation is valid: \[ p(z)=(1+z)^3\prod_{j=1}^{(N-3)/2}[1+z^2-2zγ_j], \] where $γ_j=1-2ν_j^2$ with $\{ν_j\}_{j=1}^{(N-3)/2}$ being the collection of positive roots of the equation $U'_{N-2}(x)=0$; here \[ U_j(x)=U_j(\cos t)=\frac{\sin(j+1)t}{\sin t}=2^j x^j+\ldots \] are Chebyshev polynomials of the second kind and $U'_j(x)$ are their derivatives. Similar factorization formulas are also provided for $q(z)$. As an application of the obtained results, we give the factorization formulas for the derivative of the Fejér polynomial, as well as construct certain univalent polynomials related to the polynomials $p(z)$ and $q(z)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_14009 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Some properties of the quadrinomials $p(z)=1+κ(z+z^{N-1})+z^N$ and $q(z)=1+κ(z-z^{N-1})-z^N$ Dmitrishin, Dmitriy Stokolos, Alexander Classical Analysis and ODEs 26C10 We show that all the zeros of the quadrinomial $p(z)=1+κ(z+z^{N-1})+z^N$ lie on the unit circle if and only if the inequalities \[ -1\leκ\le 1\; (\mbox{ if $N$ is even}),\;\; -1\leκ\le N/(N-2)\; (\mbox{ if $N$ is odd}) \] hold. For the quadrinomial $q(z)=1+κ(z-z^{N-1})-z^N$, the corresponding inequalities are \[ -N/(N-2)\leκ\le 1\; (\text{ if $N$ is odd}),\;\; -N/(N-2)\leκ\le N/(N-2)\; (\text{ if $N$ is even}). \] In the cases of limiting values of the parameter $κ$, we provide factorization formulas for the corresponding quadrinomials. For example, when $N$ is odd and $κ=N/(N-2)$, the following representation is valid: \[ p(z)=(1+z)^3\prod_{j=1}^{(N-3)/2}[1+z^2-2zγ_j], \] where $γ_j=1-2ν_j^2$ with $\{ν_j\}_{j=1}^{(N-3)/2}$ being the collection of positive roots of the equation $U'_{N-2}(x)=0$; here \[ U_j(x)=U_j(\cos t)=\frac{\sin(j+1)t}{\sin t}=2^j x^j+\ldots \] are Chebyshev polynomials of the second kind and $U'_j(x)$ are their derivatives. Similar factorization formulas are also provided for $q(z)$. As an application of the obtained results, we give the factorization formulas for the derivative of the Fejér polynomial, as well as construct certain univalent polynomials related to the polynomials $p(z)$ and $q(z)$. |
| title | Some properties of the quadrinomials $p(z)=1+κ(z+z^{N-1})+z^N$ and $q(z)=1+κ(z-z^{N-1})-z^N$ |
| topic | Classical Analysis and ODEs 26C10 |
| url | https://arxiv.org/abs/2410.14009 |