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Generalized continued fraction
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:
where all symbols are integers. A convenient notation is
The successive convergents are formed in a similar way to those of continued fractions. If all signs are positive,
If we write xn = pn / qn, then
(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
Then if and integer n0 such that n > n0 implies sn > ε, then the limits are equal and the continued fraction has a definite value.
Generalized continued fractions and series
is equal to the continued fraction
is equal to
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 .
- 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.
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