Difference between revisions of "Value of polygamma at 1/2"
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(Created page with "==Theorem== The following formula holds for $m=1,2,3,\ldots$: $$\psi^{(m)} \left( \dfrac{1}{2} \right) = (-1)^{m+1} m! \left( 2^{m+1}-1 \right) \zeta(m+1),$$ where $\psi^{(m)}...") |
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==References== | ==References== | ||
| − | * {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Value of polygamma at positive integer|next=}}: 6.4.4 | + | * {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Value of polygamma at positive integer|next=Value of polygamma at positive integer plus 1/2}}: 6.4.4 |
[[Category:Theorem]] | [[Category:Theorem]] | ||
[[Category:Unproven]] | [[Category:Unproven]] | ||
Revision as of 19:43, 11 June 2016
Theorem
The following formula holds for $m=1,2,3,\ldots$: $$\psi^{(m)} \left( \dfrac{1}{2} \right) = (-1)^{m+1} m! \left( 2^{m+1}-1 \right) \zeta(m+1),$$ 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): 6.4.4
