Difference between revisions of "Value of derivative of trigamma at positive integer plus 1/2"

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

Revision as of 20:24, 11 June 2016

Theorem

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

Proof

References

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