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A short proof of the simple continued fraction expansion of \(e\). (English) Zbl 1145.11012
The author presents a streamlined version of Hermite’s proof for Euler’s simple continued fraction expansion \(e= [2,1,2,1,1, 4,1,1,6,1,\dots]\). This proof rests on three sequences of integrals which are related to the convergents of this continued fraction in a simple and surprising way which has, however, no apparent motivation. The author shows that Padé approximants to the function \(e^z\) lead naturally to these integrals, and he provides evidence that Hermite found this approach, as a byproduct of his proof of the transcendence of Euler’s number \(e\), in the course of the 1892 thesis of his student H. Padé.

11A55 Continued fractions
11J70 Continued fractions and generalizations
01A55 History of mathematics in the 19th century
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