Difference between revisions of "Polygamma series representation"
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| − | + | ==Theorem== | |
| − | + | The following formula holds: | |
$$\psi^{(m)}(z)=(-1)^{m+1} m! \displaystyle\sum_{k=0}^{\infty} \dfrac{1}{(z+k)^{m+1}},$$ | $$\psi^{(m)}(z)=(-1)^{m+1} m! \displaystyle\sum_{k=0}^{\infty} \dfrac{1}{(z+k)^{m+1}},$$ | ||
where $\psi^{(m)}$ denotes the [[polygamma]] and $m!$ denotes the [[factorial]]. | where $\psi^{(m)}$ denotes the [[polygamma]] and $m!$ denotes the [[factorial]]. | ||
| − | + | ||
| − | + | ==Proof== | |
| − | + | ||
| − | + | ==References== | |
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| + | [[Category:Theorem]] | ||
| + | [[Category:Unproven]] | ||
Latest revision as of 06:34, 11 June 2016
Theorem
The following formula holds: $$\psi^{(m)}(z)=(-1)^{m+1} m! \displaystyle\sum_{k=0}^{\infty} \dfrac{1}{(z+k)^{m+1}},$$ where $\psi^{(m)}$ denotes the polygamma and $m!$ denotes the factorial.
