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Transcendence in positive characteristic and special values of hypergeometric functions

Abstract We prove a simple transcendence criterion suitable for function field arithmetic. We apply it to show the transcendence of special values at non-zero rational arguments (or more generally, at algebraic arguments which generate extension of the rational function field with less than q places at infinity) of the entire hypergeometric functions in the function field (over 𝔽 q ) context, and to obtain a new proof of the transcendence of special values at non-natural p-adic integers of the Carlitz–Goss gamma function. We also characterize in the balanced case the algebraicity of hypergeometric functions, giving an analog of the result of F. R. Villegas, based on Beukers–Heckman results and E. Landau's method in the classical hypergeometric case.