A new generalization of the continued fraction

A. D. Bruno


Abstract: We propose a new twodimensional generalization of the algorithm for expansion of a rational number into the continued fraction. The algorithm allows to obtain the successive rational approximations to a given vector. Besides the expanded vector, the algorithm uses also two associated vectors. We show that for the multiple vectors of the extremal cubic forms h₁-h₅, the new algorithm gives the periodic expansions with minimal periods. We consider also expansions for two vectors, which do not connect with the extremal forms.