Difference between revisions of "Value of polygamma at 1"

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==Theorem==
 
==Theorem==
The following formula holds:
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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]]$, and $\zeta$ denotes the [[Riemann zeta]] function.
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where $\psi^{(m)}$ denotes the [[polygamma]], $m!$ denotes the [[factorial]], and $\zeta$ denotes the [[Riemann zeta]] function.
  
 
==Proof==
 
==Proof==
  
 
==Reference==
 
==Reference==
* {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=}}: 6.4.2
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* {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Integral representation of polygamma for Re(z) greater than 0|next=Value of polygamma at positive integer}}: $6.4.2$

Latest revision as of 22:45, 17 March 2017

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