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Vallée Constant

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Consider the problem of comparing two real numbers x and y based on their continued fraction representations. Then the mean number of iterations needed to determine if x<y or x>y is given by

<N>=3/4+(180)/(pi^4)sum_(i=1)^(infty)sum_(j=i+1)^(2i)1/(i^2j^2)
(1)
=(17)/4+(360)/(pi^4)sum_(i=1)^(infty)sum_(j=1)^(i)((-1)^i)/(i^2j^2)
(2)
=17-(60)/(pi^4)[24Li_4(1/2)-pi^2ln^22+21zeta(3)ln2+ln^42]
(3)
=1.3511315744...
(4)

(OEIS A143303; Finch 2003, p. 161), where Li_4(x) is a polylogarithm.

Let p_n=P(N>=n+1), then the first few values can be given by

p_1=sum_(i=1)^(infty)1/(i^2(i+1)^2)
(5)
=1/3pi^2-3
(6)
=0.2898681336...
(7)
p_2=sum_(i=1)^(infty)sum_(j=1)^(infty)1/((ij+1)^2(ij+i+1)^2)
(8)
=-5+2/3pi^2-2zeta(3)+2sum_(n=0)^(infty)(-1)^n(n+1)zeta(n+4)[zeta(n+2)-1]
(9)
=0.0484808014...
(10)
p_3=sum_(i=1)^(infty)sum_(j=1)^(infty)sum_(k=1)^(infty)1/((ijk+i+k)^2(ijk+ij+i+k+1)^2)
(11)
=0.0102781647....
(12)

The value

v=lim_(n->infty)p_n^(1/n)=0.1994588183...
(13)

(OEIS A143302) is then known as the Vallée constant (Finch 2003, p. 161).

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References

Finch, S. R. "Hall-Montgomery Constant." §2.33 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 205-207, 2003.Sloane, N. J. A. Sequences A143302 and A143303 in "The On-Line Encyclopedia of Integer Sequences."

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Vallée Constant

Cite this as:

Weisstein, Eric W. "Vallée Constant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ValleeConstant.html

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