In mathematics, Cahen's constant is defined as the value of an infinite series of unit fractions with alternating signs:
Here denotes Sylvester's sequence, which is defined recursively by
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 was the first to introduce it and prove its irrationality.[1]
Continued fraction expansion
The majority of naturally occurring[2] mathematical constants have no known simple patterns in their continued fraction expansions.[3] Nevertheless, the complete continued fraction expansion of Cahen's constant is known: it is
is defined by the recurrence relation
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 on that . Indeed, we have , and if holds for some , then
where we used the recursion for in the first step respectively the recursion for in the final step. 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 means, there exist constants such that the inequality has infinitely many solutions , while the inequality has at most finitely many solutions . This implies (but is not equivalent to) the fact that has irrationality measure 3, which was first observed by Duverney & Shiokawa (2020).
To give a proof, denote by the sequence of convergents to Cahen's constant (that means, ).[5]
But now it follows from and the recursion for that
for every . As a consequence, the limits
- and
(recall that ) both exist by basic properties of infinite products, which is due to the absolute convergence of . Numerically, one can check that . Thus the well-known 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.
Notes
- ^ Cahen (1891).
- ^ A number is said to be naturally occurring if it is *not* defined through its decimal or continued fraction expansion. In this sense, e.g., Euler's number is naturally occurring.
- ^ Borwein et al. (2014), p. 62.
- ^ Davison & Shallit (1991).
- ^ Sloane, N. J. A. (ed.), "Sequence A006279", The On-Line Encyclopedia of Integer Sequences, OEIS Foundation
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 (2): 119–126, doi:10.1007/BF01332350, S2CID 120003890
- Borwein, Jonathan; van der Poorten, Alf; Shallit, Jeffrey; Zudilin, Wadim (2014), Neverending Fractions: An Introduction to Continued Fractions, Australian Mathematical Society Lecture Series, vol. 23, Cambridge University Press, doi:10.1017/CBO9780511902659, ISBN 978-0-521-18649-0, MR 3468515
- Duverney, Daniel; Shiokawa, Iekata (2020), "Irrationality exponents of numbers related with Cahen's constant", Monatshefte für Mathematik, 191 (1): 53–76, doi:10.1007/s00605-019-01335-0, MR 4050109, S2CID 209968916
External links
- Weisstein, Eric W., "Cahen's Constant", MathWorld
- "The Cahen constant to 4000 digits", Plouffe's Inverter, Université du Québec à Montréal, archived from the original on March 17, 2011, retrieved 2011-03-19
- "Cahen's constant (1,000,000 digits)", Darkside communication group, retrieved 2017-12-25