Generalized cyclotomic polynomials associated with regular systems of divisors and arbitrary sets of positive integers
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910432120274944 |
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| author | Tóth, László |
| author_facet | Tóth, László |
| contents | We introduce and study the generalized cyclotomic polynomials $Φ_{A,S,n}(x)$ associated with a regular system $A$ of divisors and an arbitrary set $S$ of positive integers. We show that all of these polynomials have integer coefficients, they can be expressed as the product of certain classical cyclotomic polynomials $Φ_d(x)$ with $d\mid n$, and enjoy many other properties which are similar to the classical and unitary cases. We also point out some related Menon-type identities. One of them seems to be new even for the cyclotomic polynomials $Φ_n(x)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_01278 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generalized cyclotomic polynomials associated with regular systems of divisors and arbitrary sets of positive integers Tóth, László Number Theory Primary: 11B83, Secondary 11A05, 11A25, 11C08 We introduce and study the generalized cyclotomic polynomials $Φ_{A,S,n}(x)$ associated with a regular system $A$ of divisors and an arbitrary set $S$ of positive integers. We show that all of these polynomials have integer coefficients, they can be expressed as the product of certain classical cyclotomic polynomials $Φ_d(x)$ with $d\mid n$, and enjoy many other properties which are similar to the classical and unitary cases. We also point out some related Menon-type identities. One of them seems to be new even for the cyclotomic polynomials $Φ_n(x)$. |
| title | Generalized cyclotomic polynomials associated with regular systems of divisors and arbitrary sets of positive integers |
| topic | Number Theory Primary: 11B83, Secondary 11A05, 11A25, 11C08 |
| url | https://arxiv.org/abs/2405.01278 |