User:DavidCBryant/Generalized continued fraction

In analysis, a generalized continued fraction is a generalization of regular continued fractions in canonical form in which the partial numerators and the partial denominators can assume arbitrary real or complex values.

A generalized continued fraction is an expression of the form


 * $$x = b_0 + \cfrac{a_1}{b_1 + \cfrac{a_2}{b_2 + \cfrac{a_3}{b_3 + \cfrac{a_4}{\ddots\,}}}}$$

where the an (n &gt; 0) are the partial numerators, the bn are the partial denominators, and the leading term b0 is the so-called whole or integer part of the continued fraction.

The successive convergents of the continued fraction are formed by applying the fundamental recurrence formulas:



x_0 = b_0 \qquad x_1 = \frac{b_1b_0+a_1}{b_1}\qquad x_2 = \frac{b_2(b_1b_0+a_1) + a_2b_0}{b_2b_1 + a_2}\qquad\cdots\, $$

If the sequence of convergents {xn} approaches a limit the continued fraction is convergent and has a definite value. If the sequence of convergents never approaches a limit the continued fraction is divergent. It may diverge by oscillation (for example, the odd and even convergents may approach two different limits), or it may produce an infinite number of zero denominators.

Notation
Another convenient way to express a continued fraction is



b_0+ \frac{a_1}{b_1+}\, \frac{a_2}{b_2+}\, \frac{a_3}{b_3+}\ldots $$

Generalized continued fractions and series
The following identity is due to Euler:

a_0+a_0a_1+a_0a_1a_2+a_0a_1a_2a_3+\cdots +a_0a_1a_2\cdots a_n = \frac{a_0}{1-}\, \frac{a_1}{1+a_1-}\, \frac{a_2}{1+a_2-}\, \frac{a_3}{1+a_3-}\cdots \frac{a_{n}}{1+a_n}.$$

From this follows many other results like

\frac{1}{u_1}+ \frac{1}{u_2}+ \frac{1}{u_3}+ \cdots+ \frac{1}{u_n} = \frac{1}{u_1-}\, \frac{u_1^2}{u_1+u_2-}\, \frac{u_2^2}{u_2+u_3-}\cdots \frac{u_{n-1}^2}{u_{n-1}+u_n}.$$

and

\frac{1}{a_0}+\frac{x}{a_0a_1}+\frac{x^2}{a_0a_1a_2}+ \cdots +\frac{x^n}{a_0a_1a_2\ldots a_n} = \frac{1}{a_0-}\, \frac{a_0x}{a_1+x-}\, \frac{a_1x}{a_2+x-}\, \cdots \frac{a_{n-1}x}{a_n-x}. $$

Examples


\log(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\cdots= \frac{x}{1+}\, \frac{1^2x}{2-x+}\, \frac{2^2x}{3-2x+}\, \frac{3^2x}{4-3x+}\cdots $$



\exp(x)=1+x+\frac{x^2}{2!}+\cdots= 1+\frac{x}{1-}\, \frac{x}{x+2-}\, \frac{2x}{x+3-}\, \frac{3x}{x+4-}\, \cdots $$



\exp(x)=\frac{1}{1-}\, \frac{x}{1+}\, \frac{x}{2-}\, \frac{x}{3+}\, \frac{x}{2-}\, \frac{x}{5+}\, \frac{x}{2-}\cdots $$



\pi=3+\, \frac{1}{6+}\, \frac{9}{6+}\, \frac{25}{6+}\, \frac{49}{6+}\, \frac{81}{6+}\, \frac{121}{6+}\cdots. $$

Higher dimensions
Another meaning for generalized continued fraction would be a generalisation to higher dimensions. For example, there is a close relationship between the continued fraction for the irrational real number &alpha;, and the way lattice points in two dimensions lie to either side of the line y = &alpha;x. Therefore one can ask for something relating to lattice points in three or more dimensions. One reason to study this area is to quantify the mathematical coincidence idea; for example, for monomials in several real numbers, take the logarithmic form and consider how small it can be.

There have been numerous attempts, in fact, to construct a generalised theory. Two notable ones are those of Georges Poitou and George Szekeres.