Cahen's constant

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

$C = \sum\frac{(-1)^i}{s_i-1}=\frac11 - \frac12 + \frac16 - \frac1{42} + \frac1{1806} - \cdots\approx 0.64341054629.$

By considering these fractions in pairs, we can also view 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:

$C = \sum\frac{1}{s_{2i}}=\frac12+\frac17+\frac1{1807}+\frac1{10650056950807}+\cdots$

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 and 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

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

defined by the recurrence

$q_{n+2} = q_n^2 q_{n+1} + q_n$

then the continued fraction expansion of Cahen's constant is

$[0,1,q_0^2,q_1^2,q_2^2,\ldots]$

(Davison and Shallit 1991).

[edit] References

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: 119–126. DOI:10.1007/BF01332350.