Update to Functional Equations Reference (version 1.3)

This is a substantial update of this reference document. The most important addition is the chain and substitution rules for arithmetic difference calculus (ADC). Other additions include: more properties of the discrete power function, more properties of ADC operators, definitions of analog functions, and ranges of convergence of (some) z-transforms. I also corrected some errors that were discovered since the last version.

Grab it here.

Update: Reference for Functional Equations

In this new  version of Reference for Functional Equations I added several more z-transform pairs. I also started to add binomial transform pairs. The definition for the binomial is not consistent among different authors. I arbitrarily chose one, and later I changed it. I will probably change it again. Several typos were fixed. I am working on a system to include proofs so that the tables can be checked more easily.

Difference and Functional Equations Reference

Original image by openDemocracy.

The document below contains tables and formulas useful for working with functional equations, especially di fference 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 500 formulas in its 65 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.

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.

Version 1.3

• Added the chain and substitution rules for arithmetic differences.
• Added table of functions for reference.
• Expanded the introduction somewhat.
• Added definition for arithmetic difference analogs.
• Added rules for manipulating arithmetic difference analogs.
• Added several new entries, including several functions whose sums can be expressed as the sum of $E(x) = 1/(eix + 1)$.
• Expressed the G-function (sum of the Gamma function) as a product of known functions, and replaced its notation. The notation G(x) is now used for the Barnes G-function.

Version 1.4

• Made, as always, a few corrections.
• Added the tangent sum function. There are still many details to sort out for this and related functions (cot, sec, csc, their hyperbolic counterparts, $1/(ex+1)$, and so on), and hence these sections are still messy. These will be cleaned up as the details become clear.
• Replaced some of the statements on periodic, odd, and even functions with precise versions. The previous ones were only correct up to a periodic function.
• Added the derangement function (expressed in terms of the incomplete gamma function), as well as some related Taylor series.
• Since I included the definitions of analog functions, I discovered that the intuitive notion of analogs did not correspond to the definition. Thus, the analogs of ln x and atan x have been removed / replaced. These might re-appear if the definition of analog functions is suitably adjusted.
• Made many statements on the z-transform more precise.
• Made some notations more consistent with standard notation.

Version 1.5

• Made a small correction for the binomial law for discrete powers.