Solution of fifth-degree equations (Q735965)

The authors establish a relationship between two approaches to the solution of algebraic fifth-degree equations, namely, the Hermite-Kronecker method (based on the modular elliptic equation) and the Mellin method (based on hypergeometric series). They solve the modular equation on the base of the Semusheva--Tsikh formula [\textit{A. Yu. Semusheva} and \textit{A. K. Tsikh}, Complex analysis and differential operators. Krasnoyarsk: Krasnoyarskii Gos. Univ., 134--146 (2000)], and then they deduce a formula for a series solution of the general fifth degree equation reduced by the Tschirnhaus transform \(y^5+5y=a\) and determine the convergence domain.