Cahen's constant

The Cahen constant is a mathematical constant named after the French mathematician Eugène Cahen (1865–1941) . It is a transcendent number and is defined as the limit of an alternating series of ancestral fractions .

definition

The denominators of the stem fractions are derived from the terms of the Sylvester sequence , which are recursive ${\ displaystyle S_ {0}, S_ {1}, S_ {2}, S_ {3}, ...}$

{\ displaystyle S_ {0} = 2, \ qquad S_ {n + 1} = 1 + S_ {0} \ cdots S_ {n} = 1-S_ {n} + S_ {n} ^ {2}}

for n = 0, 1, 2, 3, ...

is defined (sequence A000058 in OEIS ). With this consequence the Cahen constant is through

{\ displaystyle C = \ sum _ {k = 0} ^ {\ infty} {\ frac {(-1) ^ {k}} {S_ {k} -1}} = 1 - {\ frac {1} { 2}} + {\ frac {1} {6}} - {\ frac {1} {42}} + {\ frac {1} {1806}} - {\ frac {1} {3263442}} + - \ cdots}

defined, that is, the Sylvester-sequence is the Pierce-development of C . The convergence of the series can be shown directly with the Leibniz criterion .

properties

After combining two members of the series, you get a series whose members are only positive fractions:

{\ displaystyle C = \ sum _ {k = 0} ^ {\ infty} {\ frac {1} {S_ {2k}}} = {\ frac {1} {2}} + {\ frac {1} { 7}} + {\ frac {1} {1807}} + {\ frac {1} {10650056950807}} + \ cdots}

This representation is also provided by the Greedy algorithm for the fraction breakdown of C (the denominators form the sequence A123180 in OEIS ). The series converges rapidly because of the double exponential growth of the Sylvester sequence, each added summand quadruples the number of valid digits.

An approximation of the Cahen constant is

{\ displaystyle C = 0 {,} 64341 {\ text {}} 05462 {\ text {}} 88338 {\ text {}} 02618 {\ text {}} 22543 {\ text {}} 07757 {\ text {} } 56476 {\ text {}} 32865 {\ text {}} 87860 {\ text {}} 26823 {\ text {}} ...}

(Follow A118227 in OEIS ).

In 1891 Eugène Cahen proved in an elementary way that C is irrational (this already follows from the fact that the Pierce development does not stop). J. Les Davison and Jeffrey Shallit showed in 1991 that C is transcendent. Their proof shows more generally for all numbers whose continued fraction expansions satisfy certain simple recursive formation laws that they are transcendent. Especially for C the continued fraction expansion is through

{\ displaystyle [0; 1, q_ {0} ^ {2}, q_ {1} ^ {2}, q_ {2} ^ {2}, q_ {3} ^ {2}, ...] = [ 0; 1,1,1,4,9, ...]}

(Follow A006280 in OEIS )

given, the sequence being recursively through ${\ displaystyle q_ {0}, q_ {1}, q_ {2}, q_ {3}, ...}$

{\ displaystyle q_ {0} = q_ {1} = 1, \ qquad q_ {n + 2} = q_ {n} ^ {2} q_ {n + 1} + q_ {n}}

for n = 0, 1, 2, 3, ...

is defined (sequence A006279 in OEIS ).

For variations of the defining series of C , it is known that

{\ displaystyle C = {\ frac {1} {2}} {\ biggl (} \! 1+ \ sum _ {k = 0} ^ {\ infty} {\ frac {(-1) ^ {k}} {S_ {k}}} {\ biggr)},}

while and is still open what can be said about (the New Year's Eve sequence in this case is the Engel development , so the limit value is irrational in any case). ${\ displaystyle \ textstyle \ sum _ {k = 0} ^ {\ infty} 1 / S_ {k} = 1}$ ${\ displaystyle \ textstyle \ sum _ {k = 0} ^ {\ infty} 1 / (S_ {k} -1) = 1 {,} 69103 {\ text {}} 02067 {\ text {}} 57253 {\ text {}} 97443 {\ text {}} ...}$

literature

Steven R. Finch: Cahen's constant . In: Mathematical constants . Cambridge University Press, Cambridge 2003, ISBN 0-521-81805-2 , Chapter 6.7, pp. 434–436 (English)

Web links

Eric W. Weisstein : Cahen's Constant . In: MathWorld (English).
The Cahen constant to 4000 digits ( Memento from March 4, 2012 in the Internet Archive ) at Plouffe's Inverter (English)

Individual evidence

↑ E. Cahen : Note sur un développement des quantités numériques, qui présente quelque analogie avec celui en fractions continues . Nouvelles Annales de Mathématiques 10, 1891, pp. 508-514 (French)

↑ J. Les Davison, JO Shallit : Continued fractions for some alternating series (October 17, 1990) , months booklet for mathematics 111, 1991, pp. 119–126, doi: 10.1007 / BF01332350 (English)

↑ Finch: Cahen's constant , 2003, p. 436

[1] E. Cahen : Note sur un développement des quantités numériques, qui présente quelque analogie avec celui en fractions continues . Nouvelles Annales de Mathématiques 10, 1891, pp. 508-514 (French)