Y combinator in 140byt.es

annotated.js

function(
  // we take Haskell's fixed point combinator
  // λf.(λx.f(xx))(λx.f(xx))
  // and add an additional argument (_ stands for arguments of function)
  // λf.(λx_.f(xx)_)(λx_.f(xx)_)
  // abstract further
  // λf.(λg.gg)(λx_.f(xx)_)
  // curry the last bit
  // λf.(λg.gg)(λx.λ_.f(xx)_)
  // and alpha-rename variables (note there is no collision with the c's)
  // λa.(λc.cc)(λc.λ_.a(cc)_)
  // or more explictly
  // λa.(λc.cc)(λc.λ"arguments".a(cc)"arguments")

  a,                                // λa.
     // a function that calls its argument as if it was itself, e.g., the identity function(I){return function(n){return n==0?0:I(n-1)+1}}
  b  // undefined placeholder for pure functions or this. for the recursive function a
  ){

  return(                           // (
      function(c){                  // λc.
        return c(c)                 // cc
      }
  )                                 // )
  (                                 // (
    function(c){                    // λc.
    return function(){              // λ_.
        return a(c(c)).             // a(cc)
               apply(b,arguments)   // _
      }
    }
  )                                 // )
}

index.js

function(a,b){return(function(c){return c(c)})(function(c){return function(){return a(c(c)).apply(b,arguments)}})}

LICENSE.txt

            DO WHAT THE FUCK YOU WANT TO PUBLIC LICENSE
                    Version 2, December 2004

 Copyright (C) 2011 Peteris Erins http://peteriserins.com

 Everyone is permitted to copy and distribute verbatim or modified
 copies of this license document, and changing it is allowed as long
 as the name is changed.

            DO WHAT THE FUCK YOU WANT TO PUBLIC LICENSE
   TERMS AND CONDITIONS FOR COPYING, DISTRIBUTION AND MODIFICATION

  0. You just DO WHAT THE FUCK YOU WANT TO.

package.json

{
  "name": "ycombinator",

  "description": "A Y combinator.",

  "keywords": [
    "combinator",
    "functional",
    "lambda"
  ]
}

Would you please provide an example? I dont understand how to call this mess of functions.

If we assign this function to Y, say, then we can define the recursive identity function on integers as follows:

id = Y(function(id) { return function(n) { return n == 0 ? 0 : id(n - 1) + 1 }}

Ah, ok, now I get it. You might be able to save some bytes if you were using more imperative features. I found a version which is shorter than yours and has to be used differently:

(
  function(a){return function b(){return a.apply({},[b].concat([].slice.call(arguments)))}}
)(
  function (self, n){ return n==0? 1 : n* self( n-1);}
)( 5) // 120