## One-Ninth Constant

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

Let be Chebyshev Constants. Schönhage (1973) proved that

 (1)

It was conjectured that
 (2)

Carpenter et al. (1984) obtained
 (3)

numerically. Gonchar and Rakhmanov (1980) showed that the limit exists and disproved the 1/9 conjecture, showing that is given by
 (4)

where is the complete Elliptic Integral of the First Kind, and is the Parameter which solves
 (5)

and is the complete Elliptic Integral of the Second Kind. This gives the value for computed by Carpenter et al. (1984) is also given by the unique Positive Root of
 (6)

where
 (7)

and
 (8)

(Gonchar and Rakhmanov 1980). may also be computed by writing as
 (9)

where and , then
 (10)

(Gonchar 1990). Yet another equation for is due to Magnus (1986). is the unique solution with of
 (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. is sometimes called 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 ' 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 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.

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 über .'' J. Approx. Th. 7, 395-398, 1973.

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