Problem #2

Evaluate the exact closed-form value of the infinite series

$$S = \sum_{n=1}^{\infty} \frac{H_{n+\frac{1}{2}}}{(2n-1)^2(2n+1)}.$$

For any real $x > 0$, the generalized harmonic number is defined by

$$H_x = \psi(x+1) + \gamma,$$

where $\psi(x)$ is the digamma function and $\gamma$ is the Euler--Mascheroni constant. Here $n \in \mathbb{N}$ with $n \geq 1$, all logarithms are natural logarithms, and the series is absolutely convergent.

Determine the exact value of $S$ in closed form. Your answer must be expressed as a linear combination of rational multiples of $1$, $\pi^2$, $\ln 2$, and $\zeta(3)$.