Cubic Polynomials, Linear Shifts, and Ramanujan Simple Cubics

Abstract

We show that every monic polynomial of degree three with complex coefficients and no repeated roots is either a (vertical and horizontal) translation of y = x 3 or can be composed with a linear function to obtain a Ramanujan cubic. As a result, we gain some new insights into the roots of cubic polynomials.

A generalized computation procedure for Ramanujan-type identities and cubic Shevelev sum

“…To construct cosine Ramanujan-type identities by using Theorem 2.5, we choose a suitable 𝑟 ∈ C. For a more general result, we can use Corollary 2.7 and choose any 𝑟 ∈ C and 𝐴 ∈ R. Dresden et al [4] gave an example similar to the following example. We use our approach in the following example.…”