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: | + | 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]]. | ||
| Line 6: | Line 6: | ||
==References== | ==References== | ||
| − | * {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Value of polygamma at 1|next=}}: 6.4.3 | + | * {{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
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): 6.4.3
