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One-Ninth Constant
The one-ninth constant 
, also called the Halphen constant (Finch 2003, p. 261),
is defined by
 |
(1)
|
where 
are Chebyshev constants. Schönhage (1973)
proved that
 |
(2)
|
It was conjectured that 
, hence the name. Its reciprocal 
is sometimes called Varga's
constant. In 1981, N. Trefethen (Trefethen and Gutknecht 1983) refuted the
conjecture that 
by computing Varga's constant as
 |
(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
 |
(4)
|
(OEIS A072558) numerically.
Gonchar and Rakhmanov (1987) proved that the limit exists and disproved the 
conjecture. Gonchar presented the result at the International Congress of Mathematicians
in Berkeley, California, in 1986 (Gonchar 1990). Magnus (1986, 1988) showed that
is exactly given by
![Lambda=exp[-(piK(sqrt(1-c^2)))/(K(c))],](https://mathworld.wolfram.com/images/equations/One-NinthConstant/NumberedEquation5.svg) |
(5)
|
where 
is the complete elliptic integral
of the first kind and 
(OEIS A086199)
is the elliptic modulus in 
which solves
 |
(6)
|
and 
is the complete elliptic
integral of the second kind.
is also given by the unique root in 
of
 |
(7)
|
where
 |
(8)
|
and
 |
(9)
|
(Gonchar and Rakhmanov 1987). 
may also be computed from the prime
factorization 
,
where the 
are distinct odd primes, 
, and 
, as
 |
(10)
|
(Gonchar and Rakhmanov 1987).
A generating function for 
is given by
 |  | ![-sum_(n=1)^(infty)((-q)^n)/([1+(-q)^n]^2)](https://mathworld.wolfram.com/images/equations/One-NinthConstant/Inline22.svg) |
(11)
|
 |  |  |
(12)
|
 |  | ![1/8[1-(2/pi)^2[2E(k)-K(k)]K(k)]](https://mathworld.wolfram.com/images/equations/One-NinthConstant/Inline28.svg) |
(13)
|
for 
,
where 
and 
are complete elliptic integrals
of the first kind and complete
elliptic integrals of the second kind, respectively, and the elliptic
modulus 
is expressed in terms of the nome 
(M. Somos, pers. comm., Jul. 27, 2006).
Yet another equation for 
is due to Magnus (1988). 
is the unique solution with 
of
 |
(14)
|
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 ConstantExplore with Wolfram|Alpha
References
Carpenter, A. J.; Ruttan, A.; and Varga, R. S. "Extended Numerical Computations on the '
' 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 
in 
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 
." 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 the Rate
of Rational Approximation of Analytic Functions." Mat. Sbornik 134,
306-352, 1987. Reprinted in Math. USSR Sbornik 62, 305-348, 1989. https://doi.org/10.1070/SM1989v062n02ABEH003242.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 '
' 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 
über 
." 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.Referenced on Wolfram|Alpha
One-Ninth ConstantCite this as:
Weisstein, Eric W. "One-Ninth Constant." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/One-NinthConstant.html