In 2010, Kh.\ Hessami Pilehrood and T.\ Hessami Pilehrood introduced generating function identities used to obtain series accelerations for values of Dirichlet's $\beta$ function, via the Markov--Wilf--Zeilberger method. Inspired by these past results, together with related results introduced by Chu et al., we introduce a variety of hypergeometric recurrences that we have discovered experimentally, and we prove these recurrences using the WZ method, and apply these recurrences to obtain series acceleration identities, including a family of summations generalizing a series for $\frac{1}{\pi^2}$ due to Guillera, and a family of summations generalizing an accelerated series for Catalan's constant due to Lupa\c{s}. We also provide ``non-computer'' proofs of our hypergeometric transforms.