Can a JavaScript object property refer to another property of the same object? [duplicate]

Not with object literals (this has the same value during constructing of the literal that it did before-hand). But you can do

var carousel = new (function()
{
      this.$slider =  $('#carousel1 .slider');
      this.panes = this.$slider.children().length;
})();

This uses an object created from an anonymous function constructor.

Note that $slider and panes are public, so can be accessed as carousel.$slider, etc.