On series identities involving $\binom{4k}k$ and harmonic numbers

Abstract

The harmonic numbers are those $H_n=\sum_{0<k\le n}\frac1k\ (n=0,1,2,\ldots)$. In this paper we confirm over ten conjectural series identities with summands involving the binomial coefficient $\binom{4k}k$ and harmonic numbers. For example, we prove the identities $$\sum_{k=1}^\infty \frac{\binom{4k}{k}}{16^k}\left((22k^2-92k+11)H_{4k}-\frac{449k-275}{2}-\frac{85}{12k}\right)=-151-\frac{80}{3}\log{2}$$ and $$ \sum_{k=0}^\infty\frac{\binom{4k}{k}((11k^2+8k+1)(10H_{4k}-17H_{2k})+2k+18)}{(3k+1)(3k+2)16^k}=8\log2,$$ which were previously conjectured by Z.-W. Sun.