Complex series for 1/π
Many series for 1/π were discovered since the appearance of S. Ramanujan’s famous paper “Modular equations and approximation to π” published in 1914. Almost all these series involve only real numbers. Recently, in an attempt to prove a series for 1/π discovered by Z.-W. Sun, the authors found that a series for 1/π involving complex numbers is needed. In this article, we illustrate a method that would allow us to prove series of this type.
KeywordsHypergeometric series Singular moduli Lambert series
Mathematics Subject Classification (2000)11F11 11F03 11Y60 33C05 33C20
Unable to display preview. Download preview PDF.
- 7.Chan, H.H., Wan, J., Zudilin, W.: Legendre polynomials and Ramanujan-type series for 1/π. Isr. J. Math. (to appear) Google Scholar
- 8.Chudnovsky, D.V., Chudnovsky, G.V.: Approximations and complex multiplication according to Ramanujan. In: Ramanujan Revisited, Urbana-Champaign, IL, 1987, pp. 375–472. Academic Press, Boston (1988) Google Scholar
- 10.Ramanujan, S.: Modular equations and approximations to π. Q. J. Math. 45, 350–372 (1914) Google Scholar
- 11.Sato, T.: Apéry numbers and Ramanujan’s series for 1/π. Abstract of a talk presented at the annual meeting of the Mathematical Society of Japan (28–31 March 2002) Google Scholar
- 12.Sun, Z.-W.: List of conjectural series for powers of π and other constants. Preprint arXiv:1102.5649 [math.CA] (2011)