The number
is therefore known as the "one-ninth constant" or Halphen constant (Finch
2003, p. 261) and its reciprocal is sometimes known as 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

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
is exactly given by

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

(11)

where
and ,
then

(12)

(Gonchar and Rakhmanov 1987).

A generating function for is given by

(13)

(14)

(15)

where
and
are complete elliptic integrals of the first and 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

(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.

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''
and Similar Constants." In Nonlinear Numerical Methods and Rational Approximation
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