Solutions Of Difference Equations With Polynomial Coefficients

Title page
1 Introduction
1.1 A Historical Survey
1.2 Summary
1.3 Acknowledgments
2 Notations and Basic Definitions
2.1 Basics
2.2 Sequences, Functions, and Operators
2.2.1 The Falling and Rising Factorial and the GFF
2.2.2 Hypergeometric Terms
2.3 Difference Equations
3 Basic Theory About Difference Operators
3.1 Solutions of Difference Equations
3.2 Operations on Difference Operators
3.2.1 The Adjoint Operator
3.2.2 The Symmetric Product
3.2.3 The Greatest Common Right Divisor
3.2.4 The Lowest Common Left Multiple
4 Polynomial Solutions
4.1 The Basic Idea
4.2 Abramov and the Delta-Operator
4.3 Petkovsek and the Shift-Operator
4.4 Minimizing The Number Of Variables
4.5 Solving by Interpolation Techniques
4.5.1 Using Lagrange Interpolation
4.5.2 Using Newton Interpolation
4.6 Comparing the Bounds
4.7 Examples and Comparison
5 Rational Solutions
5.1 The Universal Denominator By Abramov
5.2 The Denominator Bound By van Hoeij
5.2.1 Improving The Scalar Case
5.3 Examples and Comparison
6 Hypergeometric Solutions
6.1 Petkovsek's Hyper
6.2 Van Hoeij's Singularities-Approach
6.2.1 Computing valuation growths
6.3 Examples and Comparison
6.4 Inhomogeneous Equations
6.4.1 Gosper's Algorithm
6.4.2 Generalizations of Gosper's Algorithm
6.4.3 Increasing the Order by 1 $
6.4.4 Examples
7 D'Alembertian Solutions
7.1 The Reduction Of Order
7.1.1 Inhomogeneous Equations
7.1.2 A Slight Improvement
7.2 Examples