One-Ninth Constant

N.B. A detailed on-line essay by S. Finch was the starting point for this entry.

Let $\lambda_{m,n}$ be Chebyshev Constants. Schönhage (1973) proved that

$$\lim_{n\to\infty} (\lambda_{0,n})^{1/n}={\textstyle{1\over 3}}.$$ (1)

It was conjectured that

$$\Lambda\equiv\lim_{n\to\infty} (\lambda_{n,n})^{1/n}={\textstyle{1\over 9}}.$$ (2)

Carpenter et al. (1984) obtained

$$\Lambda=0.1076539192\ldots$$ (3)

numerically. Gonchar and Rakhmanov (1980) showed that the limit exists and disproved the 1/9 conjecture, showing that $\Lambda$ is given by

$$\Lambda=\mathop{\rm exp}\nolimits \left[{-{\pi K(\sqrt{1-c^2}\,)\over K(c)}}\right],$$ (4)

where $K$ is the complete Elliptic Integral of the First Kind, and $c=0.9089085575485414\ldots$ is the Parameter which solves

$$K(k)=2E(k),$$ (5)

and $E$ is the complete Elliptic Integral of the Second Kind. This gives the value for $\Lambda$ computed by Carpenter et al. (1984) $\Lambda$ is also given by the unique Positive Root of

$$f(z)={\textstyle{1\over 8}},$$ (6)

where

$$f(z)\equiv \sum_{j=1}^\infty a_j z^j$$ (7)

and

$$a_j=\left\vert{\sum_{d\vert j} (-1)^d d}\right\vert$$ (8)

(Gonchar and Rakhmanov 1980). $a_j$ may also be computed by writing $j$ as

$$j=2^m {p_1}^{m_1} {p_2}^{m_2} \cdots {p_k}^{m_k},$$ (9)

where $m\geq 0$ and $m_i\geq 1$, then

$$a_j=\vert 2^{m+1}-3\vert{{p_1}^{m_1+1}-1\over p_1-1}{{p_2}^{m_2+1}-1\over p_2-1}\cdots{{p_k}^{m_k+1}-1\over p_k-1}$$ (10)

(Gonchar 1990). Yet another equation for $\Lambda$ is due to Magnus (1986). $\Lambda$ is the unique solution with $x\in(0,1)$ of

$$\sum_{k=0}^\infty (2k+1)^2(-x)^{k(k+1)/2}=0,$$ (11)

an equation which had been studied and whose root had been computed by Halphen (1886). It has therefore been suggested (Varga 1990) that the constant be called the Halphen Constant. $1/\Lambda$ is sometimes called Varga's Constant.

See also Chebyshev Constants, Halphen Constant, Varga's Constant

References

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/onenin/onenin.html

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, Fla., 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.

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.

Magnus, A. P. ``On Freud's Equations for Exponential Weights, Papers Dedicated to the Memory of Géza Freud.'' J. Approx. Th. 46, 65-99, 1986.

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.

Varga, R. S. Scientific Computations on Mathematical Problems and Conjectures. Philadelphia, PA: SIAM, 1990.