On series identities involving $\binom{4k}k$ and harmonic numbers
Bo Jiang, Zhi-Wei Sun
TL;DR
This paper resolves multiple conjectures by Z.-W. Sun on series identities involving the central binomial-type coefficient $\binom{4k}{k}$ and harmonic numbers $H_n$. It develops a generating-function framework using $f(x)=\sum_{k\ge0}\binom{4k}{k}x^k$, its companion $G_m(x)$, and derived functions $F_j(x)$ to handle weights $H_{jk}$, yielding integral representations that can be evaluated at a special point $\alpha=f(1/16)$ with $11\alpha^3-11\alpha^2-7\alpha-1=0$. This machinery is then used to prove four identities in Theorem 1 and a larger family in Theorem 2, all expressible in terms of $\log 2$ and rational constants, and further extended in Theorem 3 to a broad set of identities with bases such as $-256$, $128$, and $-72$, involving $\sqrt{2}$, $\sqrt{3}$, and $\sqrt{5}$ factors. Overall, the work provides a unifying approach to confirm Sun’s conjectures via analytic generating functions, Abel-type summations, and precise algebraic evaluations, highlighting a deep connection between binomial sums, harmonic numbers, and 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.