Volume 25, pp. 178-200, 2006.

On Euler's differential methods for continued fractions

Sergey Khrushchev

Abstract

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.

Full Text (PDF) [273 KB]

Key words

continued fractions, Ramanujan formulas, Laplace transform

AMS subject classifications

30B70

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