For q in half-integers excluding non-positive integers, Gamma^(n)(q) is transcendental for a positive proportion of n up to N, at least max(0, sqrt(N)-5/2)/N many.
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"Infinitely Often" Transcendence of Gamma-Function Derivatives
For q in half-integers excluding non-positive integers, Gamma^(n)(q) is transcendental for a positive proportion of n up to N, at least max(0, sqrt(N)-5/2)/N many.