Cahen's constant

In mathematics, Cahen's constant is defined as the value of an infinite series of unit fractions with alternating signs:

: $C = \sum_{i=0}^\infty \frac{(-1)^i}{s_i-1}=\frac11 - \frac12 + \frac16 - \frac1{42} + \frac1{1806} - \cdots\approx 0.643410546288...$ {{OEIS|id=A118227}}

Here $(s_i)_{i \geq 0}$ denotes Sylvester's sequence, which is defined recursively by

: $\begin{array}{l}
s_0~~~ = 2; \\
s_{i+1} = 1 + \prod_{j=0}^i s_j \text{ for } i \geq 0.
\end{array}$

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:

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

This constant is named after {{ill|Eugène Cahen|fr}} (also known for the Cahen–Mellin integral), who was the first to introduce it and prove its irrationality.{{sfnp|Cahen|1891}}

Continued fraction expansion

The majority of naturally occurringA 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 $e = \lim_{n \to \infty} \Big(1+\frac{1}{n}\Big)^n$ is naturally occurring. mathematical constants have no known simple patterns in their continued fraction expansions.{{sfnp|Borwein|van der Poorten|Shallit|Zudilin|2014|page=62}} Nevertheless, the complete continued fraction expansion of Cahen's constant $C$ is known: it is

$C = \left[a_0^2; a_1^2, a_2^2, a_3^2, a_4^2, \ldots\right] = [0;1,1,1,4,9,196,16641,\ldots]$

where the sequence of coefficients

{{bi|left=1.6|0, 1, 1, 1, 2, 3, 14, 129, 25298, 420984147, ... {{OEIS|id=A006279}}}}

is defined by the recurrence relation

$a_0 = 0,~a_1 = 1,~a_{n+2} = a_n\left(1 + a_n a_{n+1}\right)~\forall~n\in\mathbb{Z}_{\geqslant 0}.$

All the partial quotients of this expansion are squares of integers. Davison and Shallit made use of the continued fraction expansion to prove that $C$ is transcendental.{{sfnp|Davison|Shallit|1991}}

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 $n \geq 1$ that $1+a_n a_{n+1} = s_{n-1}$ . Indeed, we have $1+a_1 a_2 = 2 = s_0$ , and if $1+a_n a_{n+1} = s_{n-1}$ holds for some $n \geq 1$ , then

$1+a_{n+1}a_{n+2} = 1+a_{n+1} \cdot a_n(1+a_n a_{n+1})= 1+a_n a_{n+1} + (a_na_{n+1})^2 = s_{n-1} + (s_{n-1}-1)^2 = s_{n-1}^2-s_{n-1}+1 = s_n,$ where we used the recursion for $(a_n)_{n \geq 0}$ in the first step respectively the recursion for $(s_n)_{n \geq 0}$ in the final step. As a consequence, $a_{n+2} = a_n \cdot s_{n-1}$ holds for every $n \geq 1$ , from which it is easy to conclude that

$C = [0;1,1,1,s_0^2, s_1^2, (s_0s_2)^2, (s_1s_3)^2, (s_0s_2s_4)^2,\ldots]$ .

Best approximation order

Cahen's constant $C$ has best approximation order $q^{-3}$ . That means, there exist constants $K_1, K_2 > 0$ such that the inequality

$0 < \Big| C - \frac{p}{q} \Big| < \frac{K_1}{q^3}$ has infinitely many solutions $(p,q) \in \mathbb{Z} \times \mathbb{N}$ , while the inequality $0 < \Big| C - \frac{p}{q} \Big| < \frac{K_2}{q^3}$ has at most finitely many solutions $(p,q) \in \mathbb{Z} \times \mathbb{N}$ .

This implies (but is not equivalent to) the fact that $C$ has irrationality measure 3, which was first observed by {{harvtxt|Duverney|Shiokawa|2020}}.

To give a proof, denote by $(p_n/q_n)_{n \geq 0}$ the sequence of convergents to Cahen's constant (that means, $q_{n-1} = a_n \text{ for every } n \geq 1$ ).{{cite OEIS|A006279|mode=cs2}}

But now it follows from $a_{n+2} = a_n \cdot s_{n-1}$ and the recursion for $(s_n)_{n \geq 0}$ that

: $\frac{a_{n+2}}{a_{n+1}^2} = \frac{a_{n} \cdot s_{n-1}}{a_{n-1}^2 \cdot s_{n-2}^2} = \frac{a_n}{a_{n-1}^2} \cdot \frac{s_{n-2}^2 - s_{n-2} + 1}{s_{n-1}^2} = \frac{a_n}{a_{n-1}^2} \cdot \Big( 1 - \frac{1}{s_{n-1}} + \frac{1}{s_{n-1}^2} \Big)$

for every $n \geq 1$ . As a consequence, the limits

: $\alpha := \lim_{n \to \infty} \frac{q_{2n+1}}{q_{2n}^2} = \prod_{n=0}^\infty \Big( 1 - \frac{1}{s_{2n}} + \frac{1}{s_{2n}^2}\Big)$ and $\beta := \lim_{n \to \infty} \frac{q_{2n+2}}{q_{2n+1}^2} = 2 \cdot \prod_{n=0}^\infty \Big( 1 - \frac{1}{s_{2n+1}} + \frac{1}{s_{2n+1}^2}\Big)$

(recall that $s_0 = 2$ ) both exist by basic properties of infinite products, which is due to the absolute convergence of $\sum_{n=0}^\infty \Big| \frac{1}{s_{n}} - \frac{1}{s_{n}^2} \Big|$ . Numerically, one can check that $0 < \alpha < 1 < \beta < 2$ . Thus the well-known inequality

: $\frac{1}{q_n(q_n + q_{n+1})} \leq \Big| C - \frac{p_n}{q_n} \Big| \leq \frac{1}{q_nq_{n+1}}$

yields

: $\Big| C - \frac{p_{2n+1}}{q_{2n+1}} \Big| \leq \frac{1}{q_{2n+1}q_{2n+2}} = \frac{1}{q_{2n+1}^3 \cdot
\frac{q_{2n+2}}{q_{2n+1}^2}} < \frac{1}{q_{2n+1}^3}$ and $\Big| C - \frac{p_n}{q_n} \Big| \geq \frac{1}{q_n(q_n + q_{n+1})} > \frac{1}{q_n(q_n + 2q_{n}^2)} \geq \frac{1}{3q_n^3}$

for all sufficiently large $n$ . Therefore $C$ has best approximation order 3 (with $K_1 = 1 \text{ and } K_2 = 1/3$ ), where we use that any solution $(p,q) \in \mathbb{Z} \times \mathbb{N}$ to

: $0 < \Big| C - \frac{p}{q} \Big| < \frac{1}{3q^3}$

is necessarily a convergent to Cahen's constant.

Notes

References

{{citation

| last = Cahen | first = Eugène

| title = Note sur un développement des quantités numériques, qui présente quelque analogie avec celui en fractions continues

| journal = Nouvelles Annales de Mathématiques

| volume = 10

| year = 1891

| pages = 508–514}}

{{citation

| title = Continued fractions for some alternating series

| journal = Monatshefte für Mathematik

| volume = 111

| year = 1991

| pages = 119–126

| doi = 10.1007/BF01332350

| issue = 2| s2cid = 120003890 }}

{{citation

| last1 = Borwein | first1 = Jonathan

| last2 = van der Poorten | first2 = Alf

| last3 = Shallit | first3 = Jeffrey | author3-link = Jeffrey Shallit

| last4 = Zudilin | first4 = Wadim | author4-link = Wadim Zudilin

| doi = 10.1017/CBO9780511902659

| isbn = 978-0-521-18649-0

| mr = 3468515

| publisher = Cambridge University Press

| series = Australian Mathematical Society Lecture Series

| title = Neverending Fractions: An Introduction to Continued Fractions

| volume = 23

| year = 2014}}

{{citation

| last1 = Duverney | first1 = Daniel

| last2 = Shiokawa | first2 = Iekata

| doi = 10.1007/s00605-019-01335-0

| issue = 1

| journal = Monatshefte für Mathematik

| mr = 4050109

| pages = 53–76

| title = Irrationality exponents of numbers related with Cahen's constant

| volume = 191

| year = 2020| s2cid = 209968916

}}

External links

{{mathworld | title = Cahen's Constant | urlname = CahensConstant | mode=cs2}}
{{citation|title=The Cahen constant to 4000 digits |url=http://pi.lacim.uqam.ca/piDATA/cahen.txt |work=Plouffe's Inverter |publisher=Université du Québec à Montréal |accessdate=2011-03-19 |url-status=dead |archiveurl=https://web.archive.org/web/20110317202111/http://pi.lacim.uqam.ca/piDATA/cahen.txt |archivedate=March 17, 2011 }}
{{citation|title=Cahen's constant (1,000,000 digits)|url=http://ankokudan.org/d/d.htm?mathlistindex-e.html|work=Darkside communication group |accessdate=2017-12-25}}

Category:Mathematical constants

Category:Real transcendental numbers