Evaluations of some series via the WZ method

TL;DR

This paper uses the WZ method to evaluate specific series, confirming several conjectures and deriving identities involving binomial coefficients and zeta functions.

Contribution

It provides new proofs for conjectured series identities using the WZ method, expanding the toolkit for evaluating complex series.

Findings

Proved a series identity involving binomial coefficients and pi.

Confirmed a conjecture relating derivatives of gamma functions to zeta values.

Validated several previous conjectures through 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 $$\sum_{k=1}^\infty \frac{d^4}{dk^4}\left(\frac{(21k-8)\Gamma(k+1)^2}{k^3\Gamma(2k+1)}\right)=\frac{1959}2\zeta(6)-432\zeta(3)^2. $$