An identity of Andrews, multiple integrals, and very-well-poised hypergeometric series

Ramanujan Journal

Volumn 13, Issue 1-3, 2007, Pages 203-219

  Krattenthaler, C ^a,b   Rivoal, T ^c,d  

^a UNIVERSITÉ LYON 1   (France)

^b UNIVERSITY OF VIENNA   (Austria)

^c UNIVERSITÉ DE CAEN   (France)

^d INSTITUT FOURIER   (France)

Author keywords

Hypergeometric series; Riemann zeta function

Indexed keywords

EID: 33845788645     PISSN: 13824090     EISSN: None     Source Type: Journal    
DOI: 10.1007/s11139-006-0247-z     Document Type: Conference Paper

References (26)

1. Andrews, G.E.: Problems and prospects for basic hypergeometric functions. In: Askey R.A. (ed.), Theory and application of special functions. Math. Res. Center, Univ. Wisconsin, Publ. No. 35, pp. 191-224. Academic Press, New York (1975)
2. Andrews, G.E., Askey R.A., Roy, R.: Special functions. The encyclopedia of mathematics and its applications, vol. 71. Cambridge University Press, Cambridge (1999)
3. Apéry, R.: Irrationalité de ζ (2) et ζ (3). Astérisque 61, 11-13 (1979)
4. Bailey, W.N.: Generalized hypergeometric series. Cambridge University Press, Cambridge (1935)
5. Ball, K., Rivoal, T.: Irrationalité d'une infinité de valeurs de la fonction zêta aux entiers impairs. Invent. Math. 146(1), 193-207 (2001)
6. Beukers, F.: A note on the irrationality of ζ (2) and ζ (3). Bull. London Math. Soc. 11, 268-272 (1979)
7. Fischler, S.: Formes linéaires en polyzêtas et intégrales multiples. C.R. Acad. Sci. Paris, Série. I Math. 335, 1-4 (2002)
8. Gasper, G., Rahman, M.: Basic hypergeometric series. Encyclopedia of Mathematics and its Applications 35. Cambridge University Press, Cambridge (1990)
9. Koornwinder, T.: Problem in the Open Problem Session during the Workshop on "Special Functions, q-Series, and Related Topics," Toronto, Canada (1995)
10. Krattenthaler, C., Rivoal, T.: Hypergéométrie et fonction zêta de Riemann. Mem. Amer. Math. Soc., to appear. http://arXiv.org/abs/ math.NT/0311114
11. Petkovšek, M., Wilf, H.S., Zeilberger, D.: A = B, A.K. Peters, Wellesley (1996)
12. Rivoal, T.: La fonction zêta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs, C.R. Acad. Sci. Paris, Série I Math. 331(4), 267-270 (2000), http://arXiv.org/abs/math.NT/ 0008051
13. Schlosser, M.: Abel-Rothe type generalizations of Jacobi's triple product identity. In: Ismail, M.E.H. Koelink, E. (eds.), "Theory and applications of special functions. A Volume Dedicated to Mizan Rahman," Developments in Math., 13, 383-400. Kluwer Academic Publishers, Dordrecht (2005)
14. Slater, L.J.: Generalized hypergeometric functions. Cambridge University Press, Cambridge (1966)
15. 2, Mat. Sbornik 187(12), 1819-1852 (1996)
16. Sorokin, V.N.: Apéry's theorem, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 3, 48-52 (1998);
17. English translation in Moscow Univ. Math. Bull. 53.3, 48-52 (1998)
18. Vasilenko, O.N.: Certain formulae for values of the Riemann zeta function at integral points. In: Number theory and its applications. Proceedings of the science-theoretical conference (Tashkent, 1990), p. 27 (in russian)
19. Vasilyev, D.V.: Some formulas for the Riemann zeta function at integer points. Vestnik Moskov. Univ. Ser I Mat. Mekh. 1, 81-84 (1996);
20. English translation in Moscow Univ. Math. Bull. 51.1, 41-43 (1996)
21. Vasilyev, D.V.: On small linear forms for the values of the Riemann zeta-function at odd points, preprint, 1(558), Nat. Acad. Sci. Belarus, Institute Math., Minsk (2001)
22. Zeilberger, D.: Identities in search of identity. Theoret. Comput. Sci. 117, 23-38 (1993)
23. Zhao, J.: Analytic continuation of multiple zeta functions. Proc. Amer. Math. Soc. 128, 1275-1283 (2000)
24. Zlobin, S.: Integrals presented as linear forms in generalized polylogarithms. Mat. Zametki 71(5), 782-787 (2002)
25. Zudilin, W.: Well-poised hypergeometric service for diophantine problems of zeta values. J. Théor. Nombres Bordeaux 15(2), 593-626 (2003);
26. see also: Multiple integral-representations of very-well-poised hypergeometric series, e-print available at http://arXiv.org/abs/math.NT/0206177