Bifurcating Continued Fractions

Abstract

The notion of 'bifurcating continued fractions' is introduced. Two coupled sequences of non-negative integers are obtained from an ordered pair of positive real numbers in a manner that generalizes the notion of continued fractions. These sequences enable simple representations of roots of cubic equations. In particular, remarkably simple and elegant 'bifurcating continued fraction' representations of Tribonacci and Moore numbers, the cubic variations of the 'golden mean', are obtained. This is further generalized to associate m non-negative integer sequences with a set of m given real numbers so as to provide simple 'bifurcating continued fraction' representation of roots of polynomial equations of degree m+1.