The RogersRamanujan continued fraction is
defined as, with q  < 1. This
has order 5. One of its many interesting properties is that, at certain values of q, it evaluates into algebraic numbers. There are other wellknown continued fractions, of orders 4, 6, 8,
that also evaluate into algebraic numbers, but it turns out there may be such continued fractions for
ALL even orders > 2. Define two with order 20,
For example, let q = e^{π} and set, then,
There is also a nice polynomial relation between I_{1}(q) and I_{3}(q) for general q that has its basis in a modular equation, but that will be discussed in the appropriate section. Next ►
