A Galois-theoretic proof of the differential transcendence of the incomplete Gamma function

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We give simple necessary and sufficient conditions for the ∂/∂ t-transcendence of the solutions to a parameterized second order linear differential equation of the form ∂² Y/∂ x² - p ∂ Y/∂ x = 0, where p∈ F(x) is a rational function in x with coefficients in a ∂/∂ t-field F. This result is crucial for the development of an efficient algorithm to compute the parameterized Picard-Vessiot group of an arbitrary parameterized second-order linear differential equation over F(x). Our criteria imply, in particular, the ∂/∂ t-transcendence of the incomplete Gamma function γ(t,x), generalizing a result of Johnson, Reinhart, and Rubel [9].