Difference between revisions of "Series for polygamma in terms of Riemann zeta"
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(Created page with "==Theorem== The following formula holds for $|z|<1$: $$\psi^{(n)}(z+1)=(-1)^{n+1} \displaystyle\sum_{k=n}^{\infty} k! (-1)^{k+1}\zeta(k+1)z^{k-n}.$$ ==Proof== ==References==...") |
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==Theorem== | ==Theorem== | ||
The following formula holds for $|z|<1$: | The following formula holds for $|z|<1$: | ||
| − | $$\psi^{(n)}(z+1)=(-1)^{n+1} \displaystyle\sum_{k=n}^{\infty} k! (-1)^{k+1}\zeta(k+1)z^{k-n} | + | $$\psi^{(n)}(z+1)=(-1)^{n+1} \displaystyle\sum_{k=n}^{\infty} k! (-1)^{k+1}\zeta(k+1)z^{k-n},$$ |
| + | where $\psi^{(n)}$ denotes [[polygamma]] and $\zeta$ denotes [[Riemann zeta]]. | ||
==Proof== | ==Proof== | ||
Latest revision as of 22:55, 17 March 2017
Theorem
The following formula holds for $|z|<1$: $$\psi^{(n)}(z+1)=(-1)^{n+1} \displaystyle\sum_{k=n}^{\infty} k! (-1)^{k+1}\zeta(k+1)z^{k-n},$$ where $\psi^{(n)}$ denotes polygamma and $\zeta$ denotes Riemann zeta.
Proof
References
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $6.4.9$
