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:
- <math>x = \frac{b_1}{a_1\pm\frac{b_2}{a_2\pm\frac{b_3}{a_3+\,\cdots}}} <math>
where all symbols are integers. A convenient notation is
- <math>
\frac{b_1}{a_1\pm}\, \frac{b_2}{a_2\pm}\, \frac{b_3}{a_3\pm}\ldots <math>
The successive convergents are formed in a similar way to those of continued fractions. If all <math>\pm<math> signs are positive,
- <math>
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_2a_1} <math>
If we write <math>x_n=p_n/q_n<math>, then
- <math>
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}<math> (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 <math>x<math> but uniformly decrease; and all convergents of even order are less than <math>x<math> 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
- <math>
s_n= \frac{a_na_{n+1}}{b_{n+1}}. <math> Then if <math>\exists\epsilon>0<math> and integer <math>n_0<math> such that <math>n>n_0<math> implies <math>s_n>\epsilon<math>, then the limits are equal and the continued fraction has a definite value.
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Generalized continued fractions and series
The series
- <math>
\frac{1}{u_1}+ \frac{1}{u_2}+ \frac{1}{u_3}+ \cdots+ \frac{1}{u_n} <math> is equal to the continued fraction
- <math>
\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}.<math>
The series
- <math>
\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} <math> is equal to
- <math>
\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} <math>
Examples
- <math>
\log(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\ldots= \frac{x}{1+}\, \frac{1^2x}{2-x+}\, \frac{2^2x}{3-2x+}\, \frac{3^2x}{4-3x+}\ldots <math>
- <math>
\exp(x)=1+x+\frac{x^2}{2!}+\ldots= 1+\frac{x}{1-}\, \frac{x}{x+2-}\, \frac{2x}{x+3-}\, \frac{3x}{x+4-}\, \ldots <math>
- <math>
\exp(x)=\frac{1}{1-}\, \frac{z}{1+}\, \frac{z}{2-}\, \frac{z}{3+}\, \frac{z}{2-}\, \frac{z}{5+}\, \frac{z}{2-}\ldots\qquad\forall z\in C <math>
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.