Difference between revisions of "Value of polygamma at 1"
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(Created page with "==Theorem== The following formula holds: $$\psi^{(m)}(1)=(-1)^{m+1} m! \zeta(m+1),$$ where $\psi^{(m)}$ denotes the polygamma, $m!$ denotes the factorial$, and $\zeta$...") |
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==Theorem== | ==Theorem== | ||
| − | The following formula holds: | + | The following formula holds for $m=1,2,3,\ldots$: |
$$\psi^{(m)}(1)=(-1)^{m+1} m! \zeta(m+1),$$ | $$\psi^{(m)}(1)=(-1)^{m+1} m! \zeta(m+1),$$ | ||
where $\psi^{(m)}$ denotes the [[polygamma]], $m!$ denotes the [[factorial]]$, and $\zeta$ denotes the [[Riemann zeta]] function. | where $\psi^{(m)}$ denotes the [[polygamma]], $m!$ denotes the [[factorial]]$, and $\zeta$ denotes the [[Riemann zeta]] function. | ||
Revision as of 08:08, 11 June 2016
Theorem
The following formula holds for $m=1,2,3,\ldots$: $$\psi^{(m)}(1)=(-1)^{m+1} m! \zeta(m+1),$$ where $\psi^{(m)}$ denotes the polygamma, $m!$ denotes the factorial$, and $\zeta$ denotes the Riemann zeta function.
Proof
Reference
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions: 6.4.2
