Hermite’s approach to Abelian integrals revisited

In this article, we establish a new linear independence criterion for the values of certain Lauricella hypergeometric series F D with rational parameters, in both the complex and p -adic settings, over an algebraic number field. This result generalizes a theorem of C. Hermite [14] on the linear independence of certain Abelian integrals. Our proof relies on explicit Padé-type approximations to solutions of a reducible Jordan–Pochhammer differential equation, which extends the Padé approximations for certain Abelian integrals in [14]. The main novelty of our approach lies in the proof of the non-vanishing of the determinants associated with these Padé-type approximants.