[Paper Review] Identities Involving Zeros of Ramanujan and Shanks Cubic Polynomials

Stefano Barbero, Umberto Cerruti+2 more|arXiv (Cornell University)|Jan 7, 2014

Advanced Combinatorial Mathematics6 references3 citations

TL;DR

This paper establishes a deep connection between Ramanujan cubic polynomials (RCPs) and Shanks cubic polynomials (SCPs), providing an explicit trigonometric expression for the zeros of RCPs via SCPs. It proves that the zeros of RCPs are cyclically permuted under a rational transformation and derives new identities involving cosine powers and integer sequences, including a proof of a conjecture by L. E. Jefferey on sequence A198636 in the OEIS.

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 .

Motivation & Objective

• To establish a novel characterization of Ramanujan cubic polynomials (RCPs) through their connection with Shanks cubic polynomials (SCPs).
• To derive explicit trigonometric expressions for the zeros of RCPs using the zeros of SCPs.
• To demonstrate that the zeros of RCPs are cyclically permuted under a rational transformation of period three.
• To prove a conjecture by L. E. Jefferey regarding the linear recurrence of sequence A198636 in the OEIS.
• To explore connections between RCPs, Gaussian periods, and integer sequences derived from SCPs.

Proposed method

• Parameterize RCPs using two real parameters $ h $ and $ s \neq 0 $, expressing them as $ \rho(h,s,x) = x^3 + h s x^2 - (h+3)s^2 x + s^3 $.
• Use the Vieta substitution and complex root analysis to derive trigonometric expressions for the zeros of SCPs $ \rho(h,-1,x) $, involving arctangent and cosine functions.
• Define a rational transformation $ \eta_s(z) = \frac{s^2}{s - z} $, showing that it cyclically permutes the three zeros of any RCP.
• Apply the companion matrix method to show that power sums of SCP roots form linearly recurrent integer sequences.
• Use Chebyshev polynomial eigenvalues and trace identities to relate closed walks on path graphs to cosine power sums.
• Leverage known results on Gaussian periods and cyclotomic fields to derive a new identity connecting RCPs and Gaussian periods.

Experimental results

Research questions

• RQ1 How are Ramanujan cubic polynomials (RCPs) related to Shanks cubic polynomials (SCPs), and can this relationship yield a new characterization of RCPs?
• RQ2 Can the zeros of any RCP be expressed explicitly in terms of trigonometric functions using the zeros of SCPs?
• RQ3 What is the algebraic structure behind the cyclic permutation of RCP zeros under the transformation $ \eta_s(z) = \frac{s^2}{s - z} $?
• RQ4 Is the integer sequence A198636 in the OEIS a linearly recurrent sequence, and does it satisfy the conjecture by L. E. Jefferey?
• RQ5 What new identities emerge from the interplay between RCPs, SCPs, Gaussian periods, and cosine power sums?

Key findings

• The zeros of any Ramanujan cubic polynomial $ \rho(h,s,x) $ are explicitly given by $ \left\{ \alpha, \frac{s^2}{s - \alpha}, -\frac{s(s - \alpha)}{\alpha} \right\} $, forming a cyclic orbit under the transformation $ \eta_s(z) $.
• For SCPs $ \rho(h,-1,x) $, the zeros are expressed as $ \frac{1}{3}\left(h \pm 2\sqrt{\tau(h)}\cos\left(\frac{1}{3}\left(\arctan\left(\frac{3\sqrt{3}}{3+2h}\right) + k\pi\right)\right)\right) $, with $ \tau(h) = h^2 + 3h + 9 $, for $ k = 0,2,4 $.
• The sequence A198636 in the OEIS is proven to be linearly recurrent with recurrence $ A_{n+3} = 5A_{n+2} - 6A_{n+1} + A_n $, and satisfies $ A_n = 2^{2n}\left(\cos^{2n}\left(\frac{\pi}{7}\right) + \cos^{2n}\left(\frac{2\pi}{7}\right) + \cos^{2n}\left(\frac{3\pi}{7}\right)\right) $.
• A new identity is found connecting RCPs and Gaussian periods via the trace of powers of the companion matrix of SCPs.
• The power sums $ A(k,n) = \alpha^{kn} + \beta^{kn} + \gamma^{kn} $, where $ \alpha, \beta, \gamma $ are roots of $ \rho(h,-1,x) $, form linearly recurrent sequences with characteristic polynomial $ x^3 - A(k,1)x^2 + B(k,1)x - 1 $.
• When $ h = -1 $, the roots are $ 2\cos\left(\frac{2\pi}{7}\right), 2\cos\left(\frac{4\pi}{7}\right), 2\cos\left(\frac{8\pi}{7}\right) $, and the sequence $ A(2,n) $ matches A198636, confirming Jefferey’s conjecture.