Journal of Integer Sequences, Vol. 16 (2013), Article 13.8.1

Identities Involving Zeros of Ramanujan and Shanks Cubic Polynomials

Stefano Barbero, Umberto Cerruti, Nadir Murru
Department of Mathematics
University of Turin
via Carlo Alberto 10
10123 Turin

Marco Abrate
Polytechnic University of Turin
Corso Duca degli Abruzzi 24
10129 Turin


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.

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(Concerned with sequences A005471 A198632 A198636.)

Received May 6 2013; revised version received August 26 2013. Published in Journal of Integer Sequences, October 12 2013.

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