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

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(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...")
 
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==Theorem==
 
==Theorem==
The following formula holds:
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The following formula holds for $n=1,2,\ldots$:
 
$$\psi^{(m)}(n+1)=(-1)^m m! \left[ -\zeta(m+1)+1 + \dfrac{1}{2^{m+1}}+\ldots + \dfrac{1}{n^{m+1}} \right],$$
 
$$\psi^{(m)}(n+1)=(-1)^m m! \left[ -\zeta(m+1)+1 + \dfrac{1}{2^{m+1}}+\ldots + \dfrac{1}{n^{m+1}} \right],$$
 
where $\psi^{(m)}$ denotes the [[polygamma]], $m!$ denotes the [[factorial]], and $\zeta(m+1)$ denotes the [[Riemann zeta]].
 
where $\psi^{(m)}$ denotes the [[polygamma]], $m!$ denotes the [[factorial]], and $\zeta(m+1)$ denotes the [[Riemann zeta]].
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==References==
 
==References==
* {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Value of polygamma at 1|next=}}: 6.4.3
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* {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Value of polygamma at 1|next=Value of polygamma at 1/2}}: 6.4.3
  
 
[[Category:Theorem]]
 
[[Category:Theorem]]
 
[[Category:Unproven]]
 
[[Category:Unproven]]

Revision as of 19:39, 11 June 2016

Theorem

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

Proof

References