Difference between revisions of "Polygamma reflection formula"
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| − | + | ==Theorem== | |
| − | + | The following formula holds: | |
$$(-1)^m \psi^{(m)}(1-z)-\psi^{(m)}(z)=\pi \dfrac{\mathrm{d}^m}{\mathrm{d}z^m} \cot(\pi z),$$ | $$(-1)^m \psi^{(m)}(1-z)-\psi^{(m)}(z)=\pi \dfrac{\mathrm{d}^m}{\mathrm{d}z^m} \cot(\pi z),$$ | ||
where $\psi^{(m)}$ denotes the [[polygamma]], $\pi$ denotes [[pi]], and $\cot$ denotes the [[cotangent]]. | where $\psi^{(m)}$ denotes the [[polygamma]], $\pi$ denotes [[pi]], and $\cot$ denotes the [[cotangent]]. | ||
| − | + | ||
| − | + | ==Proof== | |
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| − | + | ==References== | |
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| + | [[Category:Theorem]] | ||
| + | [[Category:Unproven]] | ||
Revision as of 06:33, 11 June 2016
Theorem
The following formula holds: $$(-1)^m \psi^{(m)}(1-z)-\psi^{(m)}(z)=\pi \dfrac{\mathrm{d}^m}{\mathrm{d}z^m} \cot(\pi z),$$ where $\psi^{(m)}$ denotes the polygamma, $\pi$ denotes pi, and $\cot$ denotes the cotangent.
