In this article, we establish a new linear independence criterion for the values of certain { \it 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 \cite{ Hermite} on the linear independence of certain Abelian integrals. Our proof relies on explicit Pad\'{ e} type approximations to solutions of a reducible Jordan-Pochhammer differential equation, which extends the Pad\'{ e} approximations for certain Abelian integrals in \cite{ Hermite}. The main novelty of our approach lies in the proof of the non-vanishing of the determinants associated with these Pad\'{ e} type approximants.
Categories: math.NT