Value of derivative of trigamma at positive integer plus 1/2

Theorem

The following formula holds: $$\psi^{(m)} \prime \left( n + \dfrac{1}{2} \right)=\dfrac{\pi^2}{2} - 4 \displaystyle\sum_{k=1}^n \dfrac{1}{(2k-1)^2},$$ where $\psi^{(m)}$ denotes the polygamma and $\pi$ denotes pi.

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous): 6.4.5