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.

Identities Involving Zeros of Ramanujan and Shanks Cubic Polynomials

CERRUTI, Umberto;MURRU, NADIR
2013-01-01

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.
Vol. 16 (2013)
1
13
http://arxiv.org/pdf/1401.1474.pdf
Ramanujan Polynomials; Shanks Polynomials; Gaussian periods
Marco Abrate; Stefano Barbero; Umberto Cerruti; Nadir Murru
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2318/139373
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