In mathematics, Cahen's constant is defined as the value of the following infinite series of unit fractions with alternating signs:
. Here denotes Sylvester's sequence, which is defined recursively by and
for . 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:
Cahen's constant is named after , also known for the Cahen–Mellin integral, who was the first to introduce it and prove its irrationality.
Continued fraction expansion The majority of naturally occurring mathematical constants have no known simple patterns in their continued fraction expansions. Nevertheless, the complete continued fraction expansion of Cahen's constant is known: it is
where the sequence of coefficients
is defined for all by the recurrence relation
All the partial quotients of this expansion are squares of integers. Davison and Shallit made use of the continued fraction expansion to prove that is transcendental.
Alternatively, one may express the partial quotients in the continued fraction expansion of Cahen's constant through the terms of Sylvester's sequence. To see this, we prove by induction that . Indeed, we have , and if holds for some , then
As a consequence, holds for every , from which it is easy to conclude that .
Best approximation order Cahen's constant has best approximation order ; that is, there exist constants such that the inequality
has infinitely many solutions , while the inequality
has at most finitely many solutions . This implies that has irrationality measure 3.
To give a proof, denote by the sequence of convergents to Cahen's constant (i.e., for every ). It follows from and the recursion for that
for every . As a consequence, the limits
and
both exist by basic properties of infinite products, which is due to the absolute convergence of . Numerically, one can check that . Thus, the inequality
yields
and
for all sufficiently large . Therefore, has best approximation order 3 (with ), where we use that any solution to
is necessarily a convergent to Cahen's constant.
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