Difference between revisions of "Polygamma multiplication formula"

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==References==
 
==References==
* {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Polygamma reflection formula|next=findme}}: 6.4.8
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* {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Polygamma reflection formula|next=Series for polygamma in terms of Riemann zeta}}: $6.4.8$
  
 
[[Category:Theorem]]
 
[[Category:Theorem]]
 
[[Category:Unproven]]
 
[[Category:Unproven]]

Latest revision as of 22:51, 17 March 2017

Theorem

The following formula holds for either the pair $\delta=1, m=0$ or $\delta=0, m>0$: $$\psi^{(m)}(nz)=\delta \log(n)+\dfrac{1}{n^{m+1}} \displaystyle\sum_{k=0}^{n-1} \psi^{(n)} \left( z + \dfrac{k}{n} \right),$$ where $\psi^{(m)}$ denotes the polygamma function and $\log$ denotes the logarithm.

Proof

References