Difference between revisions of "Polygamma multiplication formula"
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(Created page with "==Theorem== The following formula holds for either the pair $\delta=1, m=0$ or $\delta=0, m>0$: $$\psi^{(m)}(nz)=\delta \log(n)+\dfrac{1}{n^{m+1}} \displaystyle\sum_{k=0}^{n-1...") |
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==References== | ==References== | ||
| − | * {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Polygamma reflection formula|next=}}: 6.4.8 | + | * {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Polygamma reflection formula|next=Series for polygamma in terms of Riemann zeta}}: $6.4.8$ |
[[Category:Theorem]] | [[Category:Theorem]] | ||
[[Category:Unproven]] | [[Category:Unproven]] | ||
Latest revision as of 22:51, 17 March 2017
Theorem
The following formula holds for either the pair $\delta=1, m=0$ or $\delta=0, m>0$: $$\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
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $6.4.8$
