Cahen's constant

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

C = ∑ i = 0 ∞ ( − 1 ) i s i − 1 = 1 1 − 1 2 + 1 6 − 1 42 + 1 1806 − ⋯ ≈ 0.643410546288... {\displaystyle C=\sum _{i=0}^{\infty }{\frac {(-1)^{i}}{s_{i}-1}}={\frac {1}{1}}-{\frac {1}{2}}+{\frac {1}{6}}-{\frac {1}{42}}+{\frac {1}{1806}}-\cdots \approx 0.643410546288...} (sequence A118227 in the OEIS)

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

s 0       = 2 ; s i + 1 = 1 + ∏ j = 0 i s j  for  i ≥ 0. {\displaystyle {\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 = ∑ 1 s 2 i = 1 2 + 1 7 + 1 1807 + 1 10650056950807 + ⋯ {\displaystyle C=\sum {\frac {1}{s_{2i}}}={\frac {1}{2}}+{\frac {1}{7}}+{\frac {1}{1807}}+{\frac {1}{10650056950807}}+\cdots }

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.[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 C {\displaystyle C} is known: it is

C = [ a 0 2 ; a 1 2 , a 2 2 , a 3 2 , a 4 2 , … ] = [ 0 ; 1 , 1 , 1 , 4 , 9 , 196 , 16641 , … ] {\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 ]}
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

a 0 = 0 ,   a 1 = 1 ,   a n + 2 = a n ( 1 + a n a n + 1 )   ∀   n ∈ Z ⩾ 0 . {\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}.}
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 {\displaystyle C} is transcendental.[4]

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 ≥ 1 {\displaystyle n\geq 1} that 1 + a n a n + 1 = s n − 1 {\displaystyle 1+a_{n}a_{n+1}=s_{n-1}} . Indeed, we have 1 + a 1 a 2 = 2 = s 0 {\displaystyle 1+a_{1}a_{2}=2=s_{0}} , and if 1 + a n a n + 1 = s n − 1 {\displaystyle 1+a_{n}a_{n+1}=s_{n-1}} holds for some n ≥ 1 {\displaystyle n\geq 1} , then

1 + a n + 1 a n + 2 = 1 + a n + 1 ⋅ a n ( 1 + a n a n + 1 ) = 1 + a n a n + 1 + ( a n a n + 1 ) 2 = s n − 1 + ( s n − 1 − 1 ) 2 = s n − 1 2 − s n − 1 + 1 = s n , {\displaystyle 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_{n}a_{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 ≥ 0 {\displaystyle (a_{n})_{n\geq 0}} in the first step respectively the recursion for ( s n ) n ≥ 0 {\displaystyle (s_{n})_{n\geq 0}} in the final step. As a consequence, a n + 2 = a n ⋅ s n − 1 {\displaystyle a_{n+2}=a_{n}\cdot s_{n-1}} holds for every n ≥ 1 {\displaystyle n\geq 1} , from which it is easy to conclude that

C = [ 0 ; 1 , 1 , 1 , s 0 2 , s 1 2 , ( s 0 s 2 ) 2 , ( s 1 s 3 ) 2 , ( s 0 s 2 s 4 ) 2 , … ] {\displaystyle C=[0;1,1,1,s_{0}^{2},s_{1}^{2},(s_{0}s_{2})^{2},(s_{1}s_{3})^{2},(s_{0}s_{2}s_{4})^{2},\ldots ]} .

Best approximation order

Cahen's constant C {\displaystyle C} has best approximation order q − 3 {\displaystyle q^{-3}} . That means, there exist constants K 1 , K 2 > 0 {\displaystyle K_{1},K_{2}>0} such that the inequality 0 < | C − p q | < K 1 q 3 {\displaystyle 0<{\Big |}C-{\frac {p}{q}}{\Big |}<{\frac {K_{1}}{q^{3}}}} has infinitely many solutions ( p , q ) ∈ Z × N {\displaystyle (p,q)\in \mathbb {Z} \times \mathbb {N} } , while the inequality 0 < | C − p q | < K 2 q 3 {\displaystyle 0<{\Big |}C-{\frac {p}{q}}{\Big |}<{\frac {K_{2}}{q^{3}}}} has at most finitely many solutions ( p , q ) ∈ Z × N {\displaystyle (p,q)\in \mathbb {Z} \times \mathbb {N} } . This implies (but is not equivalent to) the fact that C {\displaystyle C} has irrationality measure 3, which was first observed by Duverney & Shiokawa (2020).

To give a proof, denote by ( p n / q n ) n ≥ 0 {\displaystyle (p_{n}/q_{n})_{n\geq 0}} the sequence of convergents to Cahen's constant (that means, q n − 1 = a n  for every  n ≥ 1 {\displaystyle q_{n-1}=a_{n}{\text{ for every }}n\geq 1} ).[5]

But now it follows from a n + 2 = a n ⋅ s n − 1 {\displaystyle a_{n+2}=a_{n}\cdot s_{n-1}} and the recursion for ( s n ) n ≥ 0 {\displaystyle (s_{n})_{n\geq 0}} that

a n + 2 a n + 1 2 = a n ⋅ s n − 1 a n − 1 2 ⋅ s n − 2 2 = a n a n − 1 2 ⋅ s n − 2 2 − s n − 2 + 1 s n − 1 2 = a n a n − 1 2 ⋅ ( 1 − 1 s n − 1 + 1 s n − 1 2 ) {\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 )}}

for every n ≥ 1 {\displaystyle n\geq 1} . As a consequence, the limits

α := lim n → ∞ q 2 n + 1 q 2 n 2 = ∏ n = 0 ∞ ( 1 − 1 s 2 n + 1 s 2 n 2 ) {\displaystyle \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 β := lim n → ∞ q 2 n + 2 q 2 n + 1 2 = 2 ⋅ ∏ n = 0 ∞ ( 1 − 1 s 2 n + 1 + 1 s 2 n + 1 2 ) {\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 )}}

(recall that s 0 = 2 {\displaystyle s_{0}=2} ) both exist by basic properties of infinite products, which is due to the absolute convergence of ∑ n = 0 ∞ | 1 s n − 1 s n 2 | {\displaystyle \sum _{n=0}^{\infty }{\Big |}{\frac {1}{s_{n}}}-{\frac {1}{s_{n}^{2}}}{\Big |}} . Numerically, one can check that 0 < α < 1 < β < 2 {\displaystyle 0<\alpha <1<\beta <2} . Thus the well-known inequality

1 q n ( q n + q n + 1 ) ≤ | C − p n q n | ≤ 1 q n q n + 1 {\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}}}}

yields

| C − p 2 n + 1 q 2 n + 1 | ≤ 1 q 2 n + 1 q 2 n + 2 = 1 q 2 n + 1 3 ⋅ q 2 n + 2 q 2 n + 1 2 < 1 q 2 n + 1 3 {\displaystyle {\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 | C − p n q n | ≥ 1 q n ( q n + q n + 1 ) > 1 q n ( q n + 2 q n 2 ) ≥ 1 3 q n 3 {\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}}}}

for all sufficiently large n {\displaystyle n} . Therefore C {\displaystyle C} has best approximation order 3 (with K 1 = 1  and  K 2 = 1 / 3 {\displaystyle K_{1}=1{\text{ and }}K_{2}=1/3} ), where we use that any solution ( p , q ) ∈ Z × N {\displaystyle (p,q)\in \mathbb {Z} \times \mathbb {N} } to

0 < | C − p q | < 1 3 q 3 {\displaystyle 0<{\Big |}C-{\frac {p}{q}}{\Big |}<{\frac {1}{3q^{3}}}}

is necessarily a convergent to Cahen's constant.

Notes

  1. ^ Cahen (1891).
  2. ^ 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 e = lim n → ∞ ( 1 + 1 n ) n {\displaystyle e=\lim _{n\to \infty }{\Big (}1+{\frac {1}{n}}{\Big )}^{n}} is naturally occurring.
  3. ^ Borwein et al. (2014), p. 62.
  4. ^ Davison & Shallit (1991).
  5. ^ 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

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