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Generators of some Ramanujan formulas. (English) Zbl 1109.33029

Summary: We prove some Ramanujan type formulas for \(1/\pi\) but without using the theory of modular forms. Instead we use the WZ–method created by H. S. Wilf and D. Zeilberger [J. Am. Math. Soc. 3, No. 1, 147–158 (1990; Zbl 0695.05004)] and find some hypergeometric functions in two variables which are second components of WZ-pairs than can be certified using Zeilberger’s EKHAD package. These certificates have an additional property which allows us to get generalized Ramanujan’s type series which are routinely proven by computer. We call these second hypergeometric components of the WZ-pairs generators. Finding generators seems a hard task but using a kind of experimental research (explained below), we have succeeded in finding some of them. Unfortunately we have not found yet generators for the most impressive Ramanujan’s formulas. We also prove some interesting binomial sums for the constant \(1/\pi^{2}\). Finally we rewrite many of the obtained series using Pochhammer symbols and study the rate of convergence.

MSC:

33F10 Symbolic computation of special functions (Gosper and Zeilberger algorithms, etc.)
33C20 Generalized hypergeometric series, \({}_pF_q\)

Citations:

Zbl 0695.05004
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References:

[1] Bailey, W.N.: Generalized hypergeometric series. Cambridge Univ. Press (1935), p. 39 · Zbl 0011.02303
[2] Ekhad, S.B., Zeilberger, D.: A WZ proof of Ramanujan’s formula for {\(\pi\)}. In: Rassias, J.M. (ed.). Geometry, Analysis and Mechanics, World Scientific, Singapore (1994) · Zbl 0849.33003
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[4] Petkovšek, M., Wilf, H.S., Zeilberger, D.: A = B, Peters A.K.: Ltd., Appendix A (1996)
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