Hypergeometric identities for 10 extended Ramanujan-type series

By Jesús Guillera. 2008

108 citations recorded by the source corpus.

We prove, by the WZ-method, some hypergeometric identities which relate ten extended Ramanujan type series to simpler hypergeometric series. The identities we are going to prove are valid for all the values of a parameter a when they are convergent. Sometimes, even if they do not converge, they are valid if we consider these identities as limits.

Subfields: Algebra and Number Theory, Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/31af6a979e0d27ecd14a8c701fe0dcc20cec4fbc

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In Algebra and Number Theory

1. Bruce C. Berndt
2. Nayandeep Deka Baruah
3. Heng Huat Chan
4. Zhi-Wei Sun
5. George E. Andrews
6. Ken Ichi Ono
7. Eric T. Mortenson

In Applied Mathematics

1. Bruce C. Berndt
2. Nayandeep Deka Baruah

Citations in the corpus

• Open Conjectures on Congruences (2009, unknown): “1 2n 2n n nX−1 k=0 (3k + 1) 2k k 3 (−8)n−1−k ∈ Z+ for all n = 2,3,4,... . Remark. Note that a 1 = 1/4 and (2n+1)an+1+4nan = (3n+1) 2n−1 n 2 for n = 1,2,3,.... Also, the third identity of Guillera [G3] with a = 1/2 gives P∞ k=1(3k − 1)(−8)k/(k3 2k k 3 ) = −2G, where G is the Catalan constant defined”
• Eisenstein series and Ramanujan-type series for 1/π (2010, background): “Guillera [29]–[33] discovered some beautiful series for 1/π as well as for 1/π(2).”
• Divergent Ramanujan-type supercongruences (2010, unknown): “n=0 ((1) 2 )(5)n (1)(5)n (20n(2) + 8n + 1) (−1)n 22n = 8 π2 proven in [6], in which case k = 1 and z(τ1) = 1/4.”