Generators of some Ramanujan formulas |
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Authors: | Jesús Guillera |
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Institution: | (1) Av. Cesáreo Alierta 31, esc. izda 4∘A, Zaragoza, 50008, Spain |
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Abstract: | In this paper we prove some Ramanujan type formulas for 1/π but without using the theory of modular forms. Instead we use
the WZ—method created by H. Wilf and D. Zeilberger 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/π2. Finally we rewrite many of the obtained series using pochhammer symbols and study the rate of convergence.
2000 Mathematics Subject Classification Primary—33C20 |
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Keywords: | Ramanujan's series for 1/π Series for 1/π 2 Hypergeometric series WZ-method |
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