On series identities involving $\left(\genfrac{}{}{0ex}{}{4k}{k}\right)$ and harmonic numbers

Abstract

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

22 pages