# "One ninth" stuff

### Alphonse P. Magnus 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".

### But what is "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 nth power of another constant. This latter constant, long thought to be exactly 1/9, is our subject.
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 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.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...