Series involving central binomial coefficients and harmonic numbers of order 2

Abstract: We derive modular parametrizations for certain infinite series whose summands involve central binomial coefficients and second-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 $L$-values. For example, we establish the following identity that has been recently 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,] where $ \mathsf H^{(2)}_k:= \sum_{0<j\leq k}\frac{1}{j^2}$ and $ 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 $.