Difference and Functional Equations Reference


Original image by openDemocracy.

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 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.


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.

Version 1.3

  • Made several corrections.
  • 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.
  • Made some minor additions to many of the tables.
  • 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.


DiscreteCalculusTables_1_5 (PDF 4.6 MB).


  1. I believe your formulas are right. I appreciate you for the work. Yet it has a deficiency. There are too much of them. There is a summation formula for any analytical function at http://en.wikipedia.org/wiki/User:Ascoldcaves . The proof of the formula is at the site http://www.oddmaths.info/indefinitesum . Don’t bother with the proof. There is in the bottom left of the page a button “Examples”. They are live examples. To calculate a sum you have to input an analytical function at your choice and arbitrary complex boundaries and click. I think it will be funny for you. Before using examples read Instruction.
    Please tell me your opinion about it. My e-mail oddmaths@yahoo.com .

  2. Hey Ascold, thanks for your comment. I have in fact seen the sitebefore, but unfortunately could not follow the logic — most surely because of lack in my mathematical background. I have sent you a mail as well. -ht

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