Girish Mahajan (Editor)

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 = ∑ ( − 1 ) i s i − 1 = 1 1 − 1 2 + 1 6 − 1 42 + 1 1806 − ⋯ ≈ 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 = ∑ 1 s 2 i = 1 2 + 1 7 + 1 1807 + 1 10650056950807 + ⋯

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

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

defined by the recurrence relation

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 , … ]

(Davison & Shallit 1991).

References

Cahen's constant Wikipedia
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