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== | |
| − | + | ||
| − | + | ==References== | |
| + | * {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Polygamma recurrence relation|next=Polygamma multiplication formula}}: $6.4.7$ | ||
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| + | [[Category:Theorem]] | ||
| + | [[Category:Unproven]] | ||
Latest revision as of 22:46, 17 March 2017
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.
Proof
References
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $6.4.7$
