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

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].

8 pages

References in corpus (5)

• Parameterized Picard-Vessiot extensions and Atiyah extensions
• Galois Theory of Parameterized Differential Equations and Linear Differential Algebraic Groups
• Reductive linear differential algebraic groups and the Galois groups of parameterized linear differential equations
• Existence of $\partial$-parameterized Picard-Vessiot extensions over fields with algebraically closed constants
• Computing the differential Galois group of a one-parameter family of second order linear differential equations

Cited by in corpus (6)

• Reductive linear differential algebraic groups and the Galois groups of parameterized linear differential equations
• Isomonodromic differential equations and differential categories
• Calculating differential Galois groups of parametrized differential equations, with applications to hypertranscendence
• Computing the differential Galois group of a parameterized second-order linear differential equation
• Difference algebraic relations among solutions of linear differential equations
• Computation of the unipotent radical of the differential Galois group for a parameterized second-order linear differential equation