Difference and Functional Equations Reference

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The document below contains tables and formulas useful for working with functional equations, especially difference equations, and to a lesser extent, quotient equations (where differences are replaced by quotients).

The reference contains tables for forward differences, (indefinite) sums, quotients, and products. There is also a table of z-transforms, binomial transfroms, formulas for converting certain kinds of functional equations to difference equations and some discrete Taylor series. There are more than 400 formulas in its 48 pages.

This is a work in progress, so be sure to read the preface (which highlights some of the issues with this document). If you find any errors, please comment below.

Changes

Version 1.1

Added Exponential Sums to differences and sums.
Additions to the z-transform table.
Added Binomial Transform pairs.

Version 1.2

Expanded the section on the discrete power functions.
Expanded the section that explain the sue of constants in the table.
Added forms involving the following expressions to the sum (x + h) tables:
- $ax + b$
- $x^2 + a^2$
Updated all the graphs, and added some new ones.
Reorganized slightly, and fixed some typos.
Added a few examples, explanations, and additional notations in the sum ( $x + h$ ) tables.