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EnglishIn this paper we highlight the connection between Ramanujan cubic polynomials (RCPs) and a class of polynomials, the Shanks cubic polynomials (SCPs), which generate cyclic cubic fields. In this way we provide a new characterization for RCPs and we express the zeros of any RCP in explicit form, using trigonometric functions. Moreover, we observe that a cyclic transform of period three permutes these zeros. As a consequence of these results we provide many new and beautiful identities. Finally we connect RCPs to Gaussian periods, finding a new identity, and we study some integer sequences related to SCPs.
| Titolo: | Identities Involving Zeros of Ramanujan and Shanks Cubic Polynomials | |
| Autori Riconosciuti: | ||
| Autori: | Marco Abrate; Stefano Barbero; Umberto Cerruti; Nadir Murru | |
| Data di pubblicazione: | 2013 | |
| Abstract: | In this paper we highlight the connection between Ramanujan cubic polynomials (RCPs) and a class of polynomials, the Shanks cubic polynomials (SCPs), which generate cyclic cubic fields. In this way we provide a new characterization for RCPs and we express the zeros of any RCP in explicit form, using trigonometric functions. Moreover, we observe that a cyclic transform of period three permutes these zeros. As a consequence of these results we provide many new and beautiful identities. Finally we connect RCPs to Gaussian periods, finding a new identity, and we study some integer sequences related to SCPs. | |
| Volume: | Vol. 16 (2013) | |
| Pagina iniziale: | 1 | |
| Pagina finale: | 13 | |
| URL: | http://arxiv.org/pdf/1401.1474.pdf | |
| Parole Chiave: | Ramanujan Polynomials; Shanks Polynomials; Gaussian periods | |
| Rivista: | JOURNAL OF INTEGER SEQUENCES | |
| Appare nelle tipologie: | 03A-Articolo su Rivista |
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