1,367 research outputs found
Differential Galois Theory of Linear Difference Equations
We present a Galois theory of difference equations designed to measure the
differential dependencies among solutions of linear difference equations. With
this we are able to reprove Hoelder's Theorem that the Gamma function satisfies
no polynomial differential equation and are able to give general results that
imply, for example, that no differential relationship holds among solutions of
certain classes of q-hypergeometric functions.Comment: 50 page
Galois Theory of Parameterized Differential Equations and Linear Differential Algebraic Groups
We present a Galois theory of parameterized linear differential equations
where the Galois groups are linear differential algebraic groups, that is,
groups of matrices whose entries are functions of the parameters and satisfy a
set of differential equations with respect to these parameters. We present the
basic constructions and results, give examples, discuss how isomonodromic
families fit into this theory and show how results from the theory of linear
differential algebraic groups may be used to classify systems of second order
linear differential equations
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