Engel series and continued fractions. (Séries de Engel et fractions continuées.) (French) Zbl 1007.11045

An Engel series \(]a_1, a_2, \ldots [\) is an expansion \(x=\sum_{n=1}^\infty 1/a_1a_2\cdots a_n\) in integers \(a_{i+1}\geq a_i\) and \(a_1\geq 2\). In this interesting and instructive paper, the authors provide a detailed analysis and explanation of algorithms and examples giving a continued fraction expansion \([b_1, b_2, \ldots ]\) as an Engel series expansion and conversely. The underlying principle is that both the convergents (truncations) of a continued fraction and the partial sums of an Engel series are given by unimodular matrices and that those matrices factor into matrix products respectively displaying the partial quotients \(b_i\) or the quotients \(a_i\).


11J70 Continued fractions and generalizations
11J81 Transcendence (general theory)
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