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Identities Involving Zeros of Ramanujan and Shanks Cubic Polynomials

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 .

preprint2014arXivOpen access
Stefano BarberoUmberto CerrutiNadir MurruMarco Abrate
math.NT
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Open access4 authors1 topic
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Stefano BarberoUmberto CerrutiNadir MurruMarco Abrate

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