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

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∂2Y∂x2−p∂Y∂x=0, where p∈F(x)p∈F(x) is a rational function in x   with coefficients in a ∂∂t-field F  . Our criteria imply, in particular, the ∂∂t-transcendence of the incomplete Gamma function γ(t,x)γ(t,x), generalizing a result of Johnson, Reinhart, and Rubel. This result is also an important part of an efficient algorithm to compute the parameterized Picard–Vessiot group of an arbitrary parameterized second-order linear differential equation over F(x)F(x).