Geodesic
Ramanujan-type formulae and irrationality measures of some multiples of $\pi$
Sbornik. Mathematics, Tome 196 (2005) no. 7, pp. 983-998 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

An explicit construction of simultaneous Padé approximations for generalized hypergeometric series and formulae for the quantities $\pi\sqrt d$, $d\in\{1,2,3,10005\}$, in terms of these series are used for estimates of irrationality measures of these multiples of $\pi$. Other possible applications are also discussed. 
@article{SM_2005_196_7_a2,
     author = {W. V. Zudilin},
     title = {Ramanujan-type formulae and irrationality measures of some multiples of~$\pi$},
     journal = {Sbornik. Mathematics},
     pages = {983--998},
     year = {2005},
     volume = {196},
     number = {7},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2005_196_7_a2/}
}
TY  - JOUR
AU  - W. V. Zudilin
TI  - Ramanujan-type formulae and irrationality measures of some multiples of $\pi$
JO  - Sbornik. Mathematics
PY  - 2005
SP  - 983
EP  - 998
VL  - 196
IS  - 7
UR  - http://geodesic.mathdoc.fr/item/SM_2005_196_7_a2/
LA  - en
ID  - SM_2005_196_7_a2
ER  - 
%0 Journal Article
%A W. V. Zudilin
%T Ramanujan-type formulae and irrationality measures of some multiples of $\pi$
%J Sbornik. Mathematics
%D 2005
%P 983-998
%V 196
%N 7
%U http://geodesic.mathdoc.fr/item/SM_2005_196_7_a2/
%G en
%F SM_2005_196_7_a2
W. V. Zudilin. Ramanujan-type formulae and irrationality measures of some multiples of $\pi$. Sbornik. Mathematics, Tome 196 (2005) no. 7, pp. 983-998. http://geodesic.mathdoc.fr/item/SM_2005_196_7_a2/

[1] Ramanujan S., “Modular equations and approximations to $\pi$”, Quart. J. Math. Oxford Ser. (2), 45 (1914), 350–372; Collected papers of Srinivasa Ramanujan, eds. G. H. Hardy et al., Cambridge Univ. Press, Cambridge, 1927, 23–39; 2nd reprinted edition, Chelsea Publ., New York, 1962

[2] Heng Huat Chan, Wen-Chin Liaw, Tan V., “Ramanujan's class invariant $\lambda_n$ and a new class of series for $1/\pi$”, J. London Math. Soc. (2), 64:1 (2001), 93–106 | DOI | MR | Zbl

[3] Guillera J., “Some binomial series obtained by the WZ-method”, Adv. Appl. Math., 29 (2002), 599–603 | DOI | MR | Zbl

[4] Chudnovsky D. V., Chudnovsky G. V., “Approximations and complex multiplication according to Ramanujan”, Ramanujan revisited (Urbana-Champaign, Ill., 1987), Academic Press, Boston, MA, 1988, 375–472 | MR

[5] Chudnovsky G. V., “Padé approximations to the generalized hypergeometric functions. I”, J. Math. Pures Appl. (9), 58:4 (1979), 445–476 | MR | Zbl

[6] Chudnovsky D. V., Chudnovsky G. V., “Transcendental methods and theta-functions”, Theta functions, Part 2 (Bowdoin, 1987), Proc. Symposia Pure Math., 49, Amer. Math. Soc., Providence, RI, 1989, 167–232 | MR

[7] Hata M., “Rational approximations to $\pi$ and some other numbers”, Acta Arith., 63:4 (1993), 335–349 | MR | Zbl

[8] Rhin G., Viola C., “On a permutation group related to $\zeta(2)$”, Acta Arith., 77:1 (1996), 23–56 | MR | Zbl

[9] Hata M., Huttner M., “Padé approximation to the logarithmic derivative of the Gauss hypergeometric function”, Analytic number theory, eds. C. Jia, K. Matsumoto, Kluwer Acad. Publ., Netherlands, 2002, 157–172 | MR | Zbl

[10] Erdélyi A., Magnus W., Oberhettinger F., Tricomi F. G., Higher transcendental functions, vol. 1, McGraw-Hill Book Co., New York, 1953

[11] Petkovšek M., Wilf H. S., Zeilberger D., $A=B$, A. K. Peters, Ltd., Wellesley, MA, 1996 | MR | Zbl

[12] Buslaev V. I., “Sootnosheniya dlya koeffitsientov i osobye tochki funktsii”, Matem. sb., 131 (173):3 (11) (1986), 357–384 | MR | Zbl

[13] Hata M., “Legendre type polynomials and irrationality measures”, J. Reine Angew. Math., 407:1 (1990), 99–125 | MR | Zbl

[14] Guillera J., Series closely related to Ramanujan formulas for $\pi$, Unpublished manuscript, 8 pages (December 2003)