Weak transcendence in positive characteristic and hypergeometric functions for function fields

We establish a criterion for weak transcendence in positive characteristic, and then apply it to study special values of hypergeometric functions for function fields, introduced by D. S. Thakur. For these functions, it is known when they are transcendental, and in the case that they are entire transcendental functions, we apply the above criterion to show the weak transcendence of their special values at nonzero algebraic arguments. More generally, we show the same result for generalized entire fractional hypergeometric functions.