Identities Involving Zeros of Ramanujan and Shanks Cubic Polynomials
Number Theory
2014-01-08 v1
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 .
Keywords
Cite
@article{arxiv.1401.1474,
title = {Identities Involving Zeros of Ramanujan and Shanks Cubic Polynomials},
author = {Stefano Barbero and Umberto Cerruti and Nadir Murru and Marco Abrate},
journal= {arXiv preprint arXiv:1401.1474},
year = {2014}
}