Hypergeometric function representation of the roots of a certain cubic equation

Abstract

The aim in this note is to obtain new hypergeometric forms for the functions (?z ? 1 ? ?z) b ± (?z ? 1 ? ?z)?b, (?z ? 1 + ?z)b ± (?z ? 1 + ?z)?b, where b is an arbitrary parameter, in terms of Gauss hypergeometric functions. An application of these results (when b =1/3) is made to obtain the hypergeometric form of the roots of the cubic equation r3 ? r + 2/3?2/3 = 0. This complements the entry in the compendium of Prudnikov et al. on page 472, entry (68) of the table, where only the middle root (either real or purely imaginary) is given in hypergeometric form.