Series involving central binomial coefficients and higher-order harmonic numbers

Abstract

We derive modular parametrizations for certain infinite series whose summands involve central binomial coefficients and higher-order harmonic numbers. When the rates of convergence are certain rational numbers, modularity allows us to reduce the corresponding series to special values of the Dirichlet $L$-functions. For example, we establish the following identities conjectured by Sun:\[\sum_{k=0}^\infty\binom{2k}{k}^3\left[ \mathsf H_{2k}^{(2)}-\frac{25}{92}\mathsf H_{ k}^{(2)} +\frac{735L_{-7}(2)-86π^{2}}{1104}\right]\frac{1}{4096^{k}}=0,\]\[\sum_{k=0}^\infty\binom{2k}k^3\left[\mathsf H_{2k}^{(3)}-\frac{43}{352}\mathsf H_k^{(3)}\right]\frac{42k+5}{4096^k}=\frac{555ζ(3)}{77π}-\frac{32G}{11},\] where ${\mathsf{H}}_k^{(r)}:={\sum }_{0<n\le k}\frac{1}{n^r}$, $L_{-7}(2):={\sum }_{n=1}^{\infty }\left(\frac{-7}{n}\right)\frac{1}{n^2}=\frac{1}{1^2}+\frac{1}{2^2}-\frac{1}{3^2}+\frac{1}{4^2}-\frac{1}{5^2}-\frac{1}{6^2}+\frac{1}{8^2}+\cdots$, $G:={\sum }_{n=0}^{\infty }\frac{(-1{)}^n}{(2n+1{)}^2}$, and $\hat{I}\P (3):={\sum }_{n=1}^{\infty }\frac{1}{n^3}$.

37 pages, 2 tables. Theorem 1.2 added