Trinomial equation: the Hypergeometric way

This paper is devoted to the analytical treatment of trinomial equations of the form \(y^n+y=x,\) where \(y\) is the unknown and \(x\in\mathbb{C}\) is a free parameter. It is well-known that, for degree \(n\geq 5,\) algebraic equations cannot be solved by radicals; nevertheless, roots are described in terms of univariate hypergeometric or elliptic functions. This classical piece of research was founded by Hermite, Kronecker, Birkeland, Mellin and Brioschi, and continued by many other Authors. The approach mostly adopted in recent and less recent papers on this subject (see [<a href="#1">1</a>,<a href="#2">2</a>] for example) requires the use of power series, following the seminal work of Lagrange [<a href="#3">3</a>]. Our intent is to revisit the trinomial equation solvers proposed by the Italian mathematician Davide Besso in the late nineteenth century, in consideration of the fact that, by exploiting computer algebra, these methods take on an applicative and not purely theoretical relevance.

Behavior of the Trinomial ArcsB(n,k,r)when0<α<1

We deal with the familyB(n,k,r)of trinomial arcs defined as the set of roots of the trinomial equationzn=αzk+(1−α), wherez=ρeiθis a complex number,nandkare two integers such that0<k<n, andαis a real number between0and1. These arcsB(n,k,r)are continuous arcs inside the unit disk, expressed in polar coordinates(ρ,θ). The question is to prove thatρ(θ)is a decreasing function, for each trinomial arcB(n,k,r).

Hausa vowels and dipthongs

Acoustic properties of Hausa vowels and diphthongs from several speakers are investigated. The results show that Hausa is best described as having a five vowel system, where the five basic vowels have the qualities of the long vowels. Long vowels are derived as double basic vowels. The qualities of the short vowels are significantly different from those of the long vowels, but these quality differences can be accounted for by an undershoot mechanism in the speech production. The diphthong /au/ is modeled, using the formant frequencies of /a/ and /u/ with an interpolation in accordance with a trinomial equation. The diphthong /ai/ is mostly realized as a long [e:] phonetically.