Power series for roots of a trinomial and Kummer-like identities for higher order hypergeometric series

Abstract

We study the trinomial equation $x^n +px +q =0$. Here $p$ and $q$ are both real and nonzero. For $n\ge3$, expressions for the roots have been published as hypergeometric series in powers of the parameter $q^{n-1}/p^n$. For the special case of the cubic ($n=3$), we employ Kummer's identities to derive alternative series solutions in powers of the discriminant $D$, and also series in powers of $1/D$. We next derive new series, in powers of $D$ and also in powers of $1/D$, for all $n\ge 3$. The resulting series suggest identities analogous to Kummer's identities, for higher order hypergeometric series.