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

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/).

MSC:

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

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