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 a_n (n > 0) are the partial numerators, the b_n 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:

$x_{0}=b_{0}\qquad x_{1}={\frac {b_{1}b_{0}+a_{1}}{b_{1}}}\qquad x_{2}={\frac {b_{2}(b_{1}b_{0}+a_{1})+a_{2}b_{0}}{b_{2}b_{1}+a_{2}}}\qquad \cdots \,$

If the sequence of convergents {x_n} 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:

$a_{0}+a_{0}a_{1}+a_{0}a_{1}a_{2}+a_{0}a_{1}a_{2}a_{3}+\cdots +a_{0}a_{1}a_{2}\cdots a_{n}={\frac {a_{0}}{1-}}\,{\frac {a_{1}}{1+a_{1}-}}\,{\frac {a_{2}}{1+a_{2}-}}\,{\frac {a_{3}}{1+a_{3}-}}\cdots {\frac {a_{n}}{1+a_{n}}}.$

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

${\frac {1}{a_{0}}}+{\frac {x}{a_{0}a_{1}}}+{\frac {x^{2}}{a_{0}a_{1}a_{2}}}+\cdots +{\frac {x^{n}}{a_{0}a_{1}a_{2}\ldots a_{n}}}={\frac {1}{a_{0}-}}\,{\frac {a_{0}x}{a_{1}+x-}}\,{\frac {a_{1}x}{a_{2}+x-}}\,\cdots {\frac {a_{n-1}x}{a_{n}-x}}.$

Examples

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

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

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

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

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 α, and the way lattice points in two dimensions lie to either side of the line y = αx. 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.

References

• William B. Jones and W.J. Thron, "Continued Fractions Analytic Theory and Applications", Addison-Wesley, 1980. (Covers both analytic theory and history).
• Lisa Lorentzen and Haakon Waadeland, "Continued Fractions with Applications", North Holland, 1992. (Covers primarily analytic theory and some arithmetic theory).
• Oskar Perron, B.G. Teubner, "Die Lehre Von Den Kettenbruchen" Band I, II, 1954.
• George Szekeres, "Multidimensional Continued Fractions." G.Ann. Univ. Sci. Budapest Eotvos Sect. Math. 13, 113-140, 1970.
• H.S. Wall, "Analytic Theory of Continued Fractions", Chelsea, 1973.

External links

• Generalized Continued Fractions, excerpt from: Domingo Gómez Morín, La Quinta Operación Arithmética, Media Aritmónica [The Fifth Arithmetical Operation, Arithmonic Mean], ISBN 980-12-1671-9.
• The first twenty pages of Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, ISBN 0-521-81805-2, contains generalized continued fractions for √2 and the golden mean.

[[Category:Complex analysis]] [[Category:Continued fractions]]