Cahen's constant

In mathematics, Cahen's constant is defined as an infinite series of unit fractions, with alternating signs, derived from Sylvester's sequence:

Combining these fractions in pairs leads to an alternative expansion of Cahen's constant as a series of positive unit fractions formed from the terms in even positions of Sylvester's sequence. This series for Cahen's constant forms its greedy Egyptian expansion:

This constant is named after Eugène Cahen (also known for the Cahen-Mellin integral), who first formulated and investigated its series (Cahen 1891).

Cahen's constant is known to be transcendental (Davison & Shallit 1991). It is notable as being one of a small number of naturally occurring transcendental numbers for which we know the complete continued fraction expansion: if we form the sequence

0, 1, 1, 1, 2, 3, 14, 129, 25298, 420984147, ... (sequence A006279 in the OEIS)

defined by the recurrence relation

then Cahen's constant has a canonical continued fraction of:

(Davison & Shallit 1991).

ReferencesEdit

  • Cahen, Eugène (1891), "Note sur un développement des quantités numériques, qui présente quelque analogie avec celui en fractions continues", Nouvelles Annales de Mathématiques, 10: 508–514
  • Davison, J. Les; Shallit, Jeffrey O. (1991), "Continued fractions for some alternating series", Monatshefte für Mathematik, 111 (2): 119–126, doi:10.1007/BF01332350

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