Experimental determination of apéry-like identities for ζ(2n + 2)

Experimental Mathematics

Volumn 15, Issue 3, 2006, Pages 281-289

  Bailey, David H ^a   Borwein, Jonathan M ^b   Bradley, David M ^c  

^a LAWRENCE BERKELEY NATIONAL LABORATORY   (United States)

^b DALHOUSIE UNIVERSITY   (Canada)

^c UNIVERSITY OF MAINE   (United States)

Author keywords

Central binomial coefficients; Hypergeometric series; Riemann zeta function; Series acceleration

Indexed keywords

EID: 33750875429     PISSN: 10586458     EISSN: 1944950X     Source Type: Journal    
DOI: 10.1080/10586458.2006.10128968     Document Type: Article

References (20)

1. Gert Almkvist and Andrew Granville. “Borwein and Bradley’s Apéry-like Formulae for ζ(4n + 3).” Experiment. Math. 8 (1999), 197-203.
2. David H. Bailey and David J. Broadhurst. “Parallel Integer Relation Detection: Techniques and Applications.” Math. Comp. 70: 236 (2000), 1719-1736.
3. Jonathan M. Borwein and David H. Bailey., Mathematics by Experiment: Plausible Reasoning in the 21st Century. Natick, MA: A K Peters, 2004.
4. Jonathan M. Borwein and Peter B. Borwein. Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: John Wiley, 1987.
5. Jonathan M. Borwein and David M. Bradley. “Searching Symbolically for Apéry-like Formulae for Values of the Riemann Zeta Function.” SIGSAM Bulletin of Algebraic and Symbolic Manipulation 30: 2 (1996), 2-7.
6. Jonathan M. Borwein and David M. Bradley. “Empirically Determined Apéry-like Formulae for ζ(4n+3),” Experiment. Math. 6: 3 (1997), 181-194.
7. Jonathan M. Borwein, David M. Bradley, David J. Broadhurst, and Petr Lisonĕk, “Special Values of Multiple Polylogarithms,” Trans. Amer. Math. Soc. 353: 3 (2001), 907-941.
8. Jonathan M. Borwein, David J. Broadhurst, and Joel Kamnitzer, “Central Binomial Sums, Multiple Clausen Values, and Zeta Values.” Experiment. Math. 10 (2001), 25-41.
9. Jonathan M. Borwein, David H. Bailey, and Roland Girgensohn. Experimentation in Mathematics: Computational Paths to Discovery, Natick, MA: A K Peters, 2004.
10. David M. Bradley. “More Apéry-like Formulae: On Representing Values of the Riemann Zeta Function by Infinite Series Damped by Central Binomial Coefficients.” Available online (http://www.umemat.maine.edu/faculty/bradley/papers/bivar5.pdf), 2002.
11. Edward B. Burger and Robert Tubbs. Making Transcendence Transparent. New York: Springer-Verlag, 2004.
12. Louis Comtet. Advanced Combinatorics: The Art of Finite and Infinite Expansion. Dordrecht: D. Reidel, 1974.
13. Helaman R. P. Ferguson, David H. Bailey, and Stephen Arno. “Analysis of PSLQ, an Integer Relation Finding Algorithm.” Math. Comp. 68: 225 (1999), 351-369.
14. Einar Hille. Analytic Function Theory, Vol. 1. New York: Chelsea, 1976.
15. Max Koecher. Letter (in German). Math. Intelligencer 2: 2 (1979/1980), 62-64.
16. A. K. Lenstra, H. W. Lenstra Jr., and L. Lovasz, “Factoring Polynomials with Rational Coefficients.” Mathematische Annalen 261 (1982), 515-534.
17. D. Leshchiner. “Some New Identities for ζ(k).” J. Number Theory 13 (1981), 355-362.
18. T. Rivoal. “Simultaneous Generation of Koecher and Almkvist-Granville’s Apéry-like Formulae.” Experiment. Math. 13 (2004), 503-508.
19. Alfred van der Poorten. “A Proof That Euler Missed… Apéry’s Proof of the Irrationality of ζ(3).” Math. Intelligencer 1: 4 (1978/79), 195-203.
20. V. V. Zudilin. “One of the Numbers ζ(5), ζ(7), ζ(9), ζ(11) Is Irrational.” (in Russian) Uspekhi Mat. Nauk 56: 4 (340) (2001), 149-150. Translation in Russian Math. Surveys 56: 4 (2001), 774-776.