Geodesic
On Euler's differential methods for continued fractions
Electronic transactions on numerical analysis, Tome 25 (2006), pp. 178-200.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: A differential method discovered by Euler is justified and applied to give simple proofs to formulas relating important continued fractions with Laplace transforms. They include Stieltjes formulas and some Ramanujan formulas. A representation for the remainder of Leibniz's series as a continued fraction is given. We also recover the original Euler's proof for the continued fraction of hyperbolic cotangent.
Classification : 30B70
Keywords: continued fractions, Ramanujan formulas, Laplace transform 
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     author = {Khrushchev, Sergey},
     title = {On {Euler's} differential methods for continued fractions},
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     year = {2006},
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Khrushchev, Sergey. On Euler's differential methods for continued fractions. Electronic transactions on numerical analysis, Tome 25 (2006), pp. 178-200. http://geodesic.mathdoc.fr/item/ETNA_2006__25__a19/