Y Combinator vs. U Combinator vs. Typical Recursion

gistfile1.txt

// Y combinator
function Y(f) {
    return (
        (function (x) {
            return f(function (v) { return x(x)(v); }); })
        (function (x) {
            return f(function (v) { return x(x)(v); }); })
    );
}

var Y_factorial = Y(function (f) {
    return function (n) {
        if (n == 0) { return 1; }
        else { return n * f(n - 1); }
    };
});


//---------------------------------------------------------------------
Y-combinator recursive

var Y = function(f) {
    return f(function(x) { 
        return Y(f)(x);
        });
};

var Y_factorial = Y(function (f) {
    return function (n) {
        if (n == 0) { return 1; }
        else { return n * f(n - 1); }
    };
});

//---------------------------------------------------------------------

// U combinator
function U(f) {
    return f(f);
};


var U_factorial = U(function (f) {
    return function (n) {
        if (n == 0) { 
            return 1; 
        }
        else { 
            return n*(f(f)(n - 1)); 
        }
    }
});


//-------------------------------------------------------------------

// factorial recursion

var factorial = function(n) {
    if(n === 0) {
        return 1;
    } else {
        return n * factorial(n-1);
    }
};

// define omega which is the simplest non-terminating lambda expression
// var omega = U(U);