Most places that talk about special functions usually consider functions like: Airy, Bessel, Chebyshev, and Lagendre, but here we will consider a different class of special funcitons. Instead of considering special functions that are the solutions of differential equations, as most of those just mentioned are, we will be considering functions such as x^{x} or the iteration of such functions. The function x^{x} is notoriously difficult to iterate, expecially in a smooth manner. For more information about continuous iteration, click on the iteration tab above.
Unlike the functions that are represented by single letters (like f or W) or perhaps 3 letter lowercase mnemonics (like exp, log, sin, cos, ...), this website uses a wide variety of two letter uppercase mnemonics. Here is a short explaination of each term, and a link to where to find more information about it.
Helm's Alternating Series of tetrates.  
a Decremented Exponential function. See also decremented exponentials.  
the Exponential Commutator. See also here.  
the Exponential Factorial function. See also exponential factorial.  
the Exponential Gamma function. See also exponential factorial.  
one of only 2 functions on this website to be special enough to have a single letter: the Hyperpower function (coined by MacDonnell (M1)), now known as the infinite tetrate. Knoebel (K1), Galidakis (G1) and Corless et.al. (C1) all use h(x) to represent this function.  
Frappier Hyperoperators. See also here.  
Robbins Hyperoperators. See also here.  
Trappmann Hyperoperators. See also here.  
Galidakis' Hyperproductlogarithm (extension of the Lambert W function).  
Devaney's Scaled Exponentials.  
Frappier's Second Tetrate.  
the Lambert W function. See also here. 
Copyright © 2010 Andrew Robbins ( 
