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One-Ninth Constant

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Let lambda_(m,n) be Chebyshev constants. Schönhage (1973) proved that

lim_(n->infty)(lambda_(0,n))^(1/n)=1/3.
(1)

It was conjectured that the number

Lambda=lim_(n->infty)(lambda_(n,n))^(1/n)=1/9.
(2)

The number Lambda is therefore known as the "one-ninth constant" or Halphen constant (Finch 2003, p. 261) and its reciprocal V=1/Lambda is sometimes known as Varga's constant. In 1981, N. Trefethen (Trefethen and Gutknecht 1983) refuted the conjecture that Lambda=1/9 by computing Varga's constant as

V approx 9.28903
(3)

(OEIS A073007). Upon making this discovery on May 23, 1981, Trefethen (then a graduate student at Stanford University) was so excited that he sent a telegram to his coauthor M. Gutknecht in Zurich saying simply "9.28903?" Carpenter et al. (1984) subsequently confirmed this result by computing

Lambda=0.1076539192...
(4)

(OEIS A072558) numerically.

Gonchar and Rakhmanov showed that the limit exists and formally disproved the 1/9 conjecture in 1986, a result which Gonchar presented at the International Congress of Mathematicians in Berkeley, California. Magnus (1986, 1988) subsequently showed that Lambda is exactly given by

Lambda=exp[-(piK(sqrt(1-c^2)))/(K(c))],
(5)

where K is the complete elliptic integral of the first kind, and

c=0.9089085575485414...
(6)

(OEIS A086199) is the parameter which solves

K(k)=2E(k),
(7)

and E is the complete elliptic integral of the second kind.

Lambda is also given by the unique positive root of

f(z)=1/8,
(8)

where

f(z)=sum_(j=1)^inftya_jz^j
(9)

and

a_j=|sum_(d|j)(-1)^dd|
(10)

(Gonchar and Rakhmanov 1987). a_j may also be computed by writing j as

j=2^mp_1^(m_1)p_2^(m_2)...p_k^(m_k),
(11)

where m>=0 and m_i>=1, then

a_j=|2^(m+1)-3|(p_1^(m_1+1)-1)/(p_1-1)(p_2^(m_2+1)-1)/(p_2-1)...(p_k^(m_k+1)-1)/(p_k-1)
(12)

(Gonchar and Rakhmanov 1987).

A generating function for f(z) is given by

f(q)=-sum_(n=1)^(infty)((-q)^n)/([1+(-q)^n]^2)
(13)
=sum_(n=1)^(infty)(nq^n)/(1-(-q)^n)
(14)
=1/8[1-(2/pi)^2[2E(k)-K(k)]K(k)],
(15)

where K(k) and E(k) are complete elliptic integrals of the first and second kind, respectively, and the elliptic modulus k is expressed in terms of the nome q (M. Somos, pers. comm., Jul. 27, 2006).

Yet another equation for Lambda is due to Magnus (1988). Lambda is the unique solution with x in (0,1) of

sum_(k=0)^infty(2k+1)^2(-x)^(k(k+1)/2)=0,
(16)

an equation which had been studied and whose root had been computed by Halphen in 1886. It has therefore been suggested (Varga 1990) that the constant be called the Halphen constant.

See also

Chebyshev Constants, Varga's Constant

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References

Carpenter, A. J.; Ruttan, A.; and Varga, R. S. "Extended Numerical Computations on the '1/9' Conjecture in Rational Approximation Theory." In Rational Approximation and Interpolation (Tampa, FL, 1983) (Ed. P. R. Graves-Morris, E. B. Saff, and R. S. Varga). New York: Springer-Verlag, pp. 383-411, 1984.Cody, W. J.; Meinardus, G.; and Varga, R. S. "Chebyshev Rational Approximations to e^(-x) in [0,+infty) and Applications to Heat-Conduction Problems." J. Approx. Th. 2, 50-65, 1969.Dunham, C. B. and Taylor, G. D. "Continuity of Best Reciprocal Polynomial Approximation on [0,infty)." J. Approx. Th. 30, 71-79, 1980.Finch, S. R. "The 'One-Ninth' Constant." §4.5 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 259-262, 2003.Gonchar, A. A. "Rational Approximations of Analytic Functions." Amer. Math. Soc. Transl. Ser. 2 147, 25-34, 1990.Gonchar, A. A. and Rakhmanov, E. A. "Equilibrium Distributions and Degree of Rational Approximation of Analytic Functions." Math. USSR Sbornik 62, 305-348, 1980.Gonchar, A. A. and Rakhmanov, E. A. "Equilibrium Distributions and the Rate of Rational Approximation of Analytic Functions." Mat. Sbornik 34, 306-352, 1987. Reprinted in Math. USSR Sbornik 62, 305-348, 1989.Magnus, A. P. "CFGT Determination of Varga's Constant '1/9'." Inst. Preprint B-1348. Belgium: Inst. Math. U.C.L., 1986.Magnus, A. P. "On the Use of the Carathéodory-Fejér Method for Investigating '1/9' and Similar Constants." In Nonlinear Numerical Methods and Rational Approximation (Wilrijk, 1987). Dordrecht, Netherlands: Reidel, pp. 105-132, 1988.Rahman, Q. I. and Schmeisser, G. "Rational Approximation to the Exponential Function." In Padé and Rational Approximation, (Proc. Internat. Sympos., Univ. South Florida, Tampa, Fla., 1976) (Ed. E. B. Saff and R. S. Varga). New York: Academic Press, pp. 189-194, 1977.Schönhage, A. "Zur rationalen Approximierbarkeit von e^(-x) über [0,infty)." J. Approx. Th. 7, 395-398, 1973.Sloane, N. J. A. Sequences A072558, A073007, and A086199 in "The On-Line Encyclopedia of Integer Sequences."Trefethen, L. N. and Gutknecht, M. H. "The Caratheodory-Fejer Method for Real Rational Approximation." SINUM 20, 420-436, 1983.Varga, R. S. Scientific Computations on Mathematical Problems and Conjectures. Philadelphia, PA: SIAM, 1990.

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One-Ninth Constant

Cite this as:

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

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