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

TL;DR

This paper proves over ten conjectured series identities involving binomial coefficients (4k)k and harmonic numbers, confirming their validity and expanding the understanding of such series.

Contribution

The paper confirms and proves multiple conjectural series identities involving binomial coefficients and harmonic numbers, previously conjectured by Z.-W. Sun.

Findings

Confirmed over ten conjectural series identities.

Established explicit formulas involving binomial coefficients and harmonic numbers.

Connected series identities to logarithmic constants.

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.