The Euler-Minding series for branched continued fractions

Abstract

In the paper “Branched continued fractions for double power series” [ J. Comput. Appl. Math. 6 (1980) 121–125] Siemaszko generalizes for branched continued fractions the formula that expresses the difference of two successive convergents of an ordinary continued fraction. However, the generalization is not yet fit to write the branched continued fraction as an Euler-Minding series for the following reason. Indeed a convergent of the branched continued fraction can be written as a partial sum of a series but different convergents are different partial sums of different series. The next convergent cannot be obtained from the previous one by adding some terms. We shall develop here another formula that overcomes this problem.