Apéry Limits: Experiments and Proofs

References

• Alladi, K., Robinson, M. L. (1980). Legendre polynomials and irrationality. J. Reine Angew. Math. 318: 137–155.
   Web of Science ®Google Scholar
• Almkvist, G., van Straten, D., Zudilin, W. (2008). Apéry limits of differential equations of order 4 and 5. In: Yui, N., Verrill, H., Doran, C. F., eds., Proceedings of the Workshop held in Banff, AB, June 3–8, 2006. Fields Institute Communications, 54. Providence, RI: American Mathematical Society/Toronto, ON: Fields Institute for Research in Mathematical Sciences, pp. 105–123.
   Google Scholar
• Amdeberhan, T., Zeilberger, D. (1998). q-Apéry irrationality proofs by q-WZ pairs. Adv. Appl. Math. 20(2): 275–283.
   Web of Science ®Google Scholar
• Apéry, R. (1979). Irrationalité de ζ(2) et ζ(3) . Astérisque. 61: 11–13.
   Google Scholar
• Berndt, B. C. (1989). Ramanujan’s Notebooks, Part II. New York: Springer-Verlag.
   Google Scholar
• Beukers, F. (1979). A note on the irrationality of ζ(2) and ζ(3). Bull. London Math. Soc. 11(3): 268–272.
   Google Scholar
• Beukers, F. (1987). Irrationality proofs using modular forms. In: Journées arithmétiques de Besançon (Besançon, 1985), Astérisque, No. 147-148. Paris: Société mathématique de France, pp. 271–283.
   Google Scholar
• Borwein, J. M., Bailey, D. H. (2008). Mathematics by Experiment: Plausible Reasoning in the 21st Century, 2nd ed. Natick, MA: A K Peters.
   Google Scholar
• Borwein, J. M., Borwein, P. B. (1987). Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley.
   Google Scholar
• Borwein, J. M., van der Poorten, A., Shallit, J., Zudilin, W. (2014). Neverending Fractions: An Introduction to Continued Fractions. Cambridge: Cambridge Univ. Press.
   Google Scholar
• Borwein, J. M., Choi, K.-K. S., Pigulla, W. (2005). Continued fractions of tails of hypergeometric series. Amer. Math. Monthly. 112(6): 493–501.
   Web of Science ®Google Scholar
• Brown, F. (2016). Irrationality proofs for zeta values, moduli spaces and dinner parties. Mosc. J. Comb. Number Theory. 6(2/3): 102–165.
   Google Scholar
• Cooper, S. (2012). Level 10 analogues of Ramanujan’s series for 1/π . J. Ramanujan Math. Soc. 27(1): 59–76.
   Web of Science ®Google Scholar
• Cuyt, A., Petersen, V. B., Verdonk, B., Waadeland, H., Jones, W. B. (2008). Handbook of Continued Fractions for Special Functions. New York: Springer-Verlag.
   Google Scholar
• Franel, J. (1894). On a question of Laisant. L’Intermédiaire des Mathématiciens. 1: 45–47.
   Google Scholar
• Jones, W. B., Thron, W. J. (1980). Continued Fractions: Analytic Theory and Applications. Reading, MA: Addison Wesley.
   Google Scholar
• Kelley, W. G., Peterson, A. C. (2001). Difference Equations: An Introduction with Applications, 2nd ed. San Diego, CA: Academic Press.
   Google Scholar
• OEIS Foundation Inc. (2020). The On-Line Encyclopedia of Integer Sequences. Available at: oeis.org
   Google Scholar
• Perlstadt, M. A. (1987). Some recurrences for sums of powers of binomial coefficients. J. Number Theory. 27(3): 304–309. DOI: https://doi.org/10.1016/0022-314X(87)90069-2.
   Web of Science ®Google Scholar
• Petkovsek, M., Wilf, H. S., Zeilberger, D. (1996). A = B. Natick, MA: A K Peters,.
   Google Scholar
• van der Poorten, A. (1979). A proof that Euler missed …Apéry’s proof of the irrationality of ζ(3) . Math. Intelligencer. 1(4): 195–203.
   Google Scholar
• Schneider, C. (2007). Apéry’s double sum is plain sailing indeed. Electron. J. Combin. 14(1): Note 5, 3pp. DOI: https://doi.org/10.37236/1006.
   Web of Science ®Google Scholar
• Stoll, M. (1997). Bounds for the length of recurrence relations for convolutions of P-recursive sequences. Eur. J. Combin. 18(6): 707–712. DOI: https://doi.org/10.1006/eujc.1996.0123.
   Web of Science ®Google Scholar
• Wall, H. S. (1948). Analytic Theory of Continued Fractions. New York: Van Nostrand. Reprinted by the American Mathematical Society, Providence RI, 2000.
   Google Scholar
• Yang, Y. (2008). Apéry limits and special values of L-functions. J. Math. Anal. Appl. 343(1): 492–513.
   Web of Science ®Google Scholar
• Zagier, D. B. (2009). Integral solutions of Apéry-like recurrence equations. In: Harnad, J., Winternitz, P., eds. Groups and Symmetries. CRM Proceedings & Lecture Notes, Vol. 47. Providence, RI: American Mathematical Society, pp. 349–366.
   Google Scholar
• Zudilin, W. (2004). Arithmetic of linear forms involving odd zeta values. J. de Théor. des Nombres Bordeaux. 16(1): 251–291. DOI: https://doi.org/10.5802/jtnb.447.
   Google Scholar
• Zudilin, W. (2007). Approximations to -, di- and tri- logarithms. J. Comput. Appl. Math. 202(2): 450–459.
   Web of Science ®Google Scholar