Hyperelliptic integrals and multiple hypergeometric series

Abstract:

We consider the complete hyperelliptic integral \[ J(a) = \int _{\alpha (a)}^{\beta (a)} {\frac {{dx}}{{\sqrt {a - {P_n}(x)} }},}\] where $a > 0$ and \[ {P_n}(x) = \sum \limits _{k = 2}^n {{\lambda _k}{x^k},}\] with ${\lambda _2} > 0,[\alpha ,\beta ]$ being the connected component of $\{ x|{P_n}(x) \leq a\}$ containing the origin. Using a recent result concerning the Taylor expansion of the $\delta$-Dirac function, we write $J(a)$ as a power series of a parameter involving a and the ${\lambda _k}$’s. We prove this series to be a sum of multiple hypergeometric series which reduces to a single term when the number of odd monomial terms in ${P_n}$ is less than or equal to one. The region of convergence is then studied and a few particular cases are detailed.