Solving algebraic equations in terms of \(\mathcal A\)-hypergeometric series.

*(English)*Zbl 0963.33010It has long been realized that the solution of certain algebraic equations, as a function of their parameters, gives rise to hypergeometric type series. For example, as noted in the beginning paragraph of the present work, in 1757 Johann Lambert expressed the roots of the trinomial equation \(x^p+x+r\) as a Gauss hypergeometric function in the parameter \(r\). Continuing this theme, the system of linear partial differential equations satisfied by the roots of the general algebraic equation of degree \(n\), due to K. Mayr [Monatsh. Math. Phys. 45, 280-313 (1937; Zbl 0016.35702)], are identified as a special case of the type \(A\) hypergeometric system of Gel’fand and collaborators. This allows (Theorem 3.2) an explicit series solution of each of the \(n\) roots of the general \(n\)th degree equation to be presented. The convergence of these series are ensured when certain explicit inequalities on the coefficients are satisfied. However, even though there are \(2^{n-1}\) distinct forms of the \(n\) series, each labelled by a subset of \(\{1,2,\dots,n-1\}\), there are concrete examples for which the inequalities do not hold in any case.

Reviewer: Peter Forrester (Parkville