Linear independence of values of hypergeometric functions and arithmetic Gevrey series

We prove new linear independence results for the values of generalized hypergeometric functions ${}_pF_q$ at several distinct algebraic points, over suitable algebraic number fields.

Our approach provides a uniform construction of Padé approximants of type II, together with a novel non-vanishing argument for generalized Wronskians of Hermite type.

This method applies uniformly across all parameter regimes.

Even in the case $p = q+1$, we extend known results from single-point to multi-points settings over general number fields, in both complex and $p$-adic settings.

When $p < q+1$, we establish linear independence results over arbitrary number fields; and for $p > q+1$, we confirm that the values do not satisfy global linear relations in the $p$-adic setting in a framework of arithmetic Gevrey series.

The results generalize and strengthen earlier works, demonstrating the flexibility of our Padé construction for families of contiguous hypergeometric functions, through a new non-vanishing proof for the determinant, that is crucial for the universality.