A class of conjectured series representations for \(1/\pi\). (English) Zbl 1163.11031

Guillera, Jesús

Exp. Math. 15, No. 4, 409-414 (2006).

Let \(B_n\) be a sequence satisfying a linear recurrence whose coefficients are third degree polynomials in \(n\). The author gives a method, related to the theory of modular functions, to find parameters \(z,a,b\) which give conjectured series representations of the type \[ \sum_{n=0}^\infty B_n z^n (a+bn) = \frac{1}{\pi}. \] The representations are found but not proved, since the method involves inferring the closed form of functions from the beginning of their power series, employing methods of experimental mathematics (e.g., N. J. A. Sloane’s “On-Line Encyclopedia of Integer Sequences”, http://www.research.att.com/~njas/sequences/).

Reviewer: Roland Girgensohn (München)

MSC:

11F03	Modular and automorphic functions
11Y55	Calculation of integer sequences
11B83	Special sequences and polynomials
11F27	Theta series; Weil representation; theta correspondences

Keywords:

Ramanujan series; series for \(1/\pi\); Domb numbers; Apéry numbers; Dedekind eta function; Jacobi theta functions

Software:

OEIS