Difference between revisions of "Value of polygamma at 1"
From specialfunctionswiki
| Line 2: | Line 2: | ||
The following formula holds for $m=1,2,3,\ldots$: | 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]] | + | where $\psi^{(m)}$ denotes the [[polygamma]], $m!$ denotes the [[factorial]], and $\zeta$ denotes the [[Riemann zeta]] function. |
==Proof== | ==Proof== | ||
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
