Generators of some Ramanujan formulas

By Jeus Guillera. 2006

127 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Applied Mathematics, General Social Sciences

View Paper: https://www.semanticscholar.org/paper/e4e5acba9e5fb15ee5b716eab0d7130c75565345

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In Algebra and Number Theory

In Applied Mathematics

In General Social Sciences

Citations in the corpus

On the (K.2) supercongruence of Van Hamme (2015, unknown): “4) The functions (see [4] or [16]) F (n, k) = (84n(2) − 56nk + 4k(2) + 52n− 12k + 5) (−1)k((1) 2)n( 1 2)n+k( 1 2 ) 2 n−k 2(1)(2)n(1)2n−k+1 and G(n, k) = 64n(2) (−1)k((1) 2 )n( 1 2 )n+k−1( 1 2) 2 n−k 2(1)(2)n(1)2n−k”
q-Analogues of two Ramanujan-type formulas for 1/π (2018, methodology): “Note that Ekhad1 and Zeilberger [1] first applied the Wilf–Zeilberger method to prove a Ramanujan-type formula for π , and Guillera [3] found WZ pairs to reprove (1) and (2), and several more Ramanujan-type series, using the same elementary method (independent of the modular theory).”
Hypergeometric identities for 10 extended Ramanujan-type series (2008, methodology): “We consider an extension with a variable a of some new series for 1/π2 obtained by the author [5–7] and get some identities.”
A q -analogue of a Ramanujan-type supercongruence involving central binomial coefficients (2018, methodology): “Zudilin [34] has proved more Ramanujan-like supercongruences by using the WZpairs borrowed from Guillera [9, 10].”
A q-microscope for supercongruences (2018, background): “Concluding remarks and open problems Since Ramanujan’s formula (1) has a WZ proof [6], it is natural to ask whether there is a q-WZ proof of its q-analogue (4).”
Eisenstein series and Ramanujan-type series for 1/π (2010, unknown): “Guillera [31] proved the identity (5.”