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arXiv · 1304.1917

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

Carlos E. Arreche
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Abstract

We give simple necessary and sufficient conditions for the $\frac{\partial}{\partial t}$-transcendence of the solutions to a parameterized second order linear differential equation of the form \frac{\partial^2 Y}{\partial x^2} - p \frac{\partial Y}{\partial x} = 0, where $p\in F(x)$ is a rational function in $x$ with coefficients in a $\frac{\partial}{\partial t}$-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 $\frac{\partial}{\partial t}$-transcendence of the incomplete Gamma function $γ(t,x)$, generalizing a result of Johnson, Reinhart, and Rubel [9].

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Carlos E. Arreche. 2013-04-06. A Galois-theoretic proof of the differential transcendence of the incomplete Gamma function. https://doi.org/10.1016/j.jalgebra.2013.04.037

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