Difference between revisions of "Polygamma multiplication formula"
From specialfunctionswiki
(Created page with "==Theorem== The following formula holds for either the pair $\delta=1, m=0$ or $\delta=0, m>0$: $$\psi^{(m)}(nz)=\delta \log(n)+\dfrac{1}{n^{m+1}} \displaystyle\sum_{k=0}^{n-1...") |
(No difference)
|
Revision as of 20:22, 11 June 2016
Theorem
The following formula holds for either the pair $\delta=1, m=0$ or $\delta=0, m>0$: $$\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
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous): 6.4.8
