Value of derivative of trigamma at positive integer plus 1/2
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Theorem
The following formula holds: $$\psi^{\prime} \left( n + \dfrac{1}{2} \right)=\dfrac{\pi^2}{2} - 4 \displaystyle\sum_{k=1}^n \dfrac{1}{(2k-1)^2},$$ where $\psi^{(m)}$ denotes the digamma and $\pi$ denotes pi.
Proof
References
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): 6.4.5
