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
Polytechnic University of Turin
Corso Duca degli Abruzzi 24
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.
Full version: pdf,
(Concerned with sequences
Received May 6 2013;
revised version received August 26 2013.
Published in Journal of Integer Sequences, October 12 2013.
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