Institut Math. Univ. Louvain

Chemin du Cyclotron 2

B-1348 Louvain-la-Neuve

Belgium

mail to alphonse.magnus@uclouvain.be

Various things that I wrote on "1/9".

The norm error of best rational approximation of degree *n*
to exp(-*x*) on the positive real line behaves like a constant
times the *n*th power of another constant. This latter constant,
long thought to be exactly 1/9, is our subject.

Start with the very interesting and complete introduction by
S. Finch at
this link. Also
here.

Oh, the material has been slightly changed since book publication,
see here the preprint of this
section.

Also in the three items

Sequences A073007, decimal expansion of 1/'1/9', A072558, decimal expansion of '1/9', A072559, continued fraction of '1/9' (pdf)

of
N. J. A. Sloane, editor (2003), The On-Line Encyclopedia of Integer Sequences,

**http://www.research.att.com/~njas/sequences/**

**
STOP PRESS ** A.I. Aptekarev just gave in

"Sharp constants for rational approximation of analytic
functions" (in Russian),
*Mathematical Sbornik*, Vol **193**(1), 2002, pp. 3-72

a proof that the error behaves like

2 times the (*n*+1/2)th power of "1/9" !

The english translation is published in
the * Sb. Math. * vol. ** 193** (2002) no. 1-2, 1-72.
MR1906170 (2003g:30070).

The English preprint (ps 2734K) (pdf 597K)

The very first announcement of the formula, October 1985. The postscript file, The pdf file.

More things, in construction.

The famous page 287 of G.H.Halphen's
Traité des fonctions elliptiques
et de leurs applications. Première partie.
Théorie des fonctions
elliptiques et de leurs développements en séries.
Gauthier-Villars, Paris, 1886.

The whole books are available at
http://gallica.bnf.fr,
also at
http://moa.cit.cornell.edu.
See also here the pages 285,
286, and
288.

ZOOM pages
285,
286,
287,
288.

A page
(disappeared??)
of the Rouen university on Halphen

A page by the Société des amis de
la bibliothèque de l'Ecole Polytechnique

A.P.MAGNUS: On the use of Carathéodory-Fejér method for
investigating "1/9" and similar constants, pp. 105-132
*in* A.CUYT, Editor:
* Nonlinear Numerical Methods and Rational Approximation*, D.Reidel,
Dordrecht 1988.

pdf file 385K

A.P.MAGNUS: Asymptotics and super asymptotics of
best rational approximation error norms
for the exponential function (the '1/9' problem)
by the Carathéodory-Fejér's method,
pp. 173-185 *in *
A. Cuyt, editor:
*Nonlinear Methods and Rational Approximation II *,
Kluwer, Dordrecht, 1994.

ps file 420K
pdf file 309K

A.P. Magnus, J. Meinguet,
The elliptic functions and integrals of the `1/9' problem,
presented at Antwerpen international conference on
rational approximation June 6--11 1999
ICRA99,
*Numerical Algorithms* **24**: (1-2) (2000) 117-139.
The ps preprint(320K).

The pdf (220K).

The PARI programs nine.gp
ninem.gp.

An essay on ninth Kings, Popes, and Symphonies in history...