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 1 connect RCPs to Gaussian periods, finding a new identity, and we study some integer sequences related to SCPs. 1 Ramanujan and Shanks polynomials Definition 1. A Ramanujan cubic polynomial (RCP) x+px+qx+r is a cubic polynomial, with p, q, r ∈ R, r 6= 0, which has real zeros and with coefficients satisfying the relation pr 1 3 + 3r 2 3 + q = 0. (1) Shevelev [6] first introduced the definition of RCP, in honour of the great mathematician Srinivasa Ramanujan, since such a class of polynomials arises from a theorem proved by Ramanujan [4]. Witula [11] gave the characterization for the RCPs showed in the next theorem. Theorem 2. All RCPs have the form: x − P (γ − 1) (γ − 1) (γ − 2) r 1 3 x − P (2− γ) (1− γ) (2− γ) r 2 3 x+ r = = (