On two conjectures due to Sun

We prove two conjectures due to Sun concerning binomial-harmonic sums. First, we introduce a proof of a formula for Catalan’s constant that had been conjectured by Sun in 2014. Then, using a similar approach as in our first proof, we solve an open problem due to Sun involving the sequence of alternating odd harmonic numbers. Our methods, more broadly, allow us to reduce difficult binomial-harmonic sums to finite combinations of dilogarithms that are evaluable using previously known algorithms.