In mathematics, Cahen's constant is defined as the value of an infinite series of unit fractions with alternating signs:
(sequence A118227 in the OEIS)
Here
denotes Sylvester's sequence, which is defined recursively by
![{\displaystyle {\begin{array}{l}s_{0}~~~=2;\\s_{i+1}=1+\prod _{j=0}^{i}s_{j}{\text{ for }}i\geq 0.\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9bc1466297d59fa16b196b37b25f4ad2a1b01bdc)
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:
![{\displaystyle C=\sum {\frac {1}{s_{2i}}}={\frac {1}{2}}+{\frac {1}{7}}+{\frac {1}{1807}}+{\frac {1}{10650056950807}}+\cdots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/fcc06a2318c69c1d90b826ed1fd3aaa3aa837fa0)
This constant is named after Eugène Cahen [fr] (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[2] 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
![{\displaystyle C=\left[a_{0}^{2};a_{1}^{2},a_{2}^{2},a_{3}^{2},a_{4}^{2},\ldots \right]=[0;1,1,1,4,9,196,16641,\ldots ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/67d797a4ab5d84b6fe15bb47209b1bbeb98c2f1a)
where the sequence of coefficients
0, 1, 1, 1, 2, 3, 14, 129, 25298, 420984147, ... (sequence
A006279 in the
OEIS)
is defined by the recurrence relation
![{\displaystyle a_{0}=0,~a_{1}=1,~a_{n+2}=a_{n}\left(1+a_{n}a_{n+1}\right)~\forall ~n\in \mathbb {Z} _{\geqslant 0}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d4c55a94980e18b77949ea6deb1b23f632ca83b)
All the partial quotients of this expansion are squares of integers. Davison and Shallit made use of the continued fraction expansion to prove that
![{\displaystyle C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029)
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 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
![{\displaystyle {\frac {a_{n+2}}{a_{n+1}^{2}}}={\frac {a_{n}\cdot s_{n-1}}{a_{n-1}^{2}\cdot s_{n-2}^{2}}}={\frac {a_{n}}{a_{n-1}^{2}}}\cdot {\frac {s_{n-2}^{2}-s_{n-2}+1}{s_{n-1}^{2}}}={\frac {a_{n}}{a_{n-1}^{2}}}\cdot {\Big (}1-{\frac {1}{s_{n-1}}}+{\frac {1}{s_{n-1}^{2}}}{\Big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/56bac7333418e3894d32c49c6181fc1a890bbecd)
for every
. As a consequence, the limits
and ![{\displaystyle \beta :=\lim _{n\to \infty }{\frac {q_{2n+2}}{q_{2n+1}^{2}}}=2\cdot \prod _{n=0}^{\infty }{\Big (}1-{\frac {1}{s_{2n+1}}}+{\frac {1}{s_{2n+1}^{2}}}{\Big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e14acabe1d70b28df74e53e4711c3db695fa79c8)
(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
![{\displaystyle {\frac {1}{q_{n}(q_{n}+q_{n+1})}}\leq {\Big |}C-{\frac {p_{n}}{q_{n}}}{\Big |}\leq {\frac {1}{q_{n}q_{n+1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/964bd17d02b57dd01f41b5dd4563dc19ddfdf481)
yields
and ![{\displaystyle {\Big |}C-{\frac {p_{n}}{q_{n}}}{\Big |}\geq {\frac {1}{q_{n}(q_{n}+q_{n+1})}}>{\frac {1}{q_{n}(q_{n}+2q_{n}^{2})}}\geq {\frac {1}{3q_{n}^{3}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/794cad9d4c119942756df3f107682e7d4d480d2b)
for all sufficiently large
. Therefore
has best approximation order 3 (with
), where we use that any solution
to
![{\displaystyle 0<{\Big |}C-{\frac {p}{q}}{\Big |}<{\frac {1}{3q^{3}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6aeccf78b8b5e8e9b0728275d9f5ef95f5f0c5cc)
is necessarily a convergent to Cahen's constant.
Notes
- ^ 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. - ^ 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