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

TL;DR

This paper provides a Galois-theoretic proof establishing the differential transcendence of the incomplete Gamma function, offering criteria that generalize previous results and aid in computing parameterized Picard-Vessiot groups.

Contribution

It introduces simple necessary and sufficient conditions for the differential transcendence of solutions to a class of parameterized second-order linear differential equations, advancing the theoretical framework.

Findings

Criteria for $rac{ ext{ extbackslash}partial}{ ext{ extbackslash}partial t}$-transcendence of solutions

Generalization of previous results on the incomplete Gamma function

Development of an algorithm for computing parameterized Picard-Vessiot groups

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 $\gamma(t,x)$, generalizing a result of Johnson, Reinhart, and Rubel [9].