Difference between revisions of "Polygamma recurrence relation"
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| − | + | ==Theorem== | |
| − | + | The following formula holds: | |
$$\psi^{(m)}(z+1)=\psi^{(m)}(z)+\dfrac{(-1)^mm!}{z^{m+1}},$$ | $$\psi^{(m)}(z+1)=\psi^{(m)}(z)+\dfrac{(-1)^mm!}{z^{m+1}},$$ | ||
where $\psi^{(m)}$ denotes the [[polygamma]] and $m!$ denotes the [[factorial]]. | where $\psi^{(m)}$ denotes the [[polygamma]] and $m!$ denotes the [[factorial]]. | ||
| − | + | ||
| − | + | ==Proof== | |
| − | + | ||
| − | + | ==References== | |
| + | * {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Value of derivative of trigamma at positive integer plus 1/2|next=Polygamma reflection formula}}: $6.4.6$ | ||
| + | |||
| + | [[Category:Theorem]] | ||
| + | [[Category:Unproven]] | ||
Latest revision as of 22:46, 17 March 2017
Theorem
The following formula holds: $$\psi^{(m)}(z+1)=\psi^{(m)}(z)+\dfrac{(-1)^mm!}{z^{m+1}},$$ where $\psi^{(m)}$ denotes the polygamma and $m!$ denotes the factorial.
Proof
References
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $6.4.6$
