Generalized continued fraction

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In mathematics, a generalized continued fraction is a generalization of the concept of continued fraction in which the numerators are allowed to differ from unity. They are useful in the theory of infinite summation of series.

A generalized continued fraction is an expression such as:

$x = \cfrac{b_1}{a_1\pm\cfrac{b_2}{a_2\pm\cfrac{b_3}{a_3+\,\cdots}}}$

where all symbols are integers. A convenient notation is

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

The successive convergents are formed in a similar way to those of continued fractions. If all $\pm$ signs are positive,

$x_1=\frac{b_1}{a_1}\qquad x_2=\frac{a_2b_1}{a_2a_1+b_2}\qquad x_3=\frac{a_3a_2b_1+b_3b_1}{a_3(a_2a_1+b_2)+b_3a_1}$

If we write $x n = p n / q n$ , then

$p_{n+1}=a_{n+1}p_n+b_{n+1}p_{n-1},\qquad q_{n+1}=a_{n+1}q_n+b_{n+1}q_{n-1}$

(if the signs are negative, replace "+" with "-" in the above formula).

If the positive sign is chosen, then (as for ordinary continued fractions) all convergents of odd order are greater than $x$ but uniformly decrease; and all convergents of even order are less than $x$ but uniformly increase.

Thus odd convergents tend to a limit, and even convergents tend to a limit. If the limits are not equal, the continued fraction is said to be oscillating. To determine whether the limits are equal, define

$s_n= \frac{a_na_{n+1}}{b_{n+1}}.$

Then if $\exists\epsilon>0$ and integer $n 0$ such that $n > n 0$ implies $s n > ε$ , then the limits are equal and the continued fraction has a definite value.

1 Generalized continued fractions and series
2 Examples
3 Higher dimensions
4 References
5 External links

[edit] 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

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

[edit] Examples

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

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

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

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

[edit] 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.

[edit] 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.

[edit] 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.

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Category: Continued fractions