Difference between revisions of "Value of polygamma at positive integer"

From specialfunctionswiki
Jump to: navigation, search
(Created page with "==Theorem== The following formula holds: $$\psi^{(m)}(n+1)=(-1)^m m! \left[ -\zeta(m+1)+1 + \dfrac{1}{2^{m+1}}+\ldots + \dfrac{1}{n^{m+1}} \right],$$ where $\psi^{(m)}$ denote...")
(No difference)

Revision as of 19:38, 11 June 2016

Theorem

The following formula holds: $$\psi^{(m)}(n+1)=(-1)^m m! \left[ -\zeta(m+1)+1 + \dfrac{1}{2^{m+1}}+\ldots + \dfrac{1}{n^{m+1}} \right],$$ where $\psi^{(m)}$ denotes the polygamma, $m!$ denotes the factorial, and $\zeta(m+1)$ denotes the Riemann zeta.

Proof

References