In earlier posts (since deleted) I discussed some of the properties of continued fractions and showed how Lambert’s continued fractions for tanh x and tan x could be derived and used to prove that these functions are always irrational numbers for non-zero, rational x and consequently that the exponential number, e, and pi are irrational.
In what follows, I simplify and generalise those discussions. The following is an outline, the detail can be found in the pdf here – revised in May 2016.
Irrationality criteria and continued fractions
Suppose f(x) is some function that we can represent in the following manner:
where the are polynomials in x with integer coefficients and has the property that it becomes rapidly smaller in absolute value for a given x as k becomes larger. We call this a continued fraction representation of f(x) and it is not too hard to derive from it the following expression
where p, q are defined recursively by
Clearly and are integral polynomials in x. If d(n) is the degree of whichever of and has the higher degree, and if x = u/v is rational with u, v being integers having greatest common factor 1, then if we multiply the above expression by then it takes the form
where k, h are integers and we can make use of the following theorem:
f(x) is irrational if there exist integers h and k such that 0 < |kf(x) – h| < ε for any given 1 > ε > 0
Lambert’s continued fractions
Suppose and are functions that are can be expressed as even, absolutely convergent Maclaurin series. Then we can obtain a sequence of equations
If we take and
then it is not too hard to show that can be expressed as a continued fraction of the form
These are Lambert’s continued fractions. The process described above can be carried through for them to show that tanh x/x and tan x/x, and hence tanh x and tan x, are irrational for non-zero rational x.