Evaluations of some series via the WZ method

Abstract

In this paper, we evaluate some series via the WZ method, and confirm several previous conjectures. For example, we prove the following two identities conjectured by the second author: $$\sum_{k=0}^{\infty} \frac{(28k^2 + 10k + 1) \binom{2k}{k}^5}{(6k + 1)(-64)^k \binom{3k}{k} \binom{6k}{3k}} = \frac{3}{\pi}$$ and $$\sum_{k=1}^\infty \frac{d^4}{dk^4}\left(\frac{(21k-8)\Gamma(k+1)^2}{k^3\Gamma(2k+1)}\right)=\frac{1959}2\zeta(6)-432\zeta(3)^2. $$