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The Rogers-Ramanujan continued fraction is defined as,


with || < 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 well-known 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 and set,


then,

There is also a nice polynomial relation between I1(q) and I3(q) for general q that has its basis in a modular equation, but that will be discussed in the appropriate section.


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