Polygamma

The polygamma function of order $m$, $\psi^{(m)}(z)$, is defined by the formula $$\psi^{(m)}(z) = \dfrac{\mathrm{d}^{m+1}}{\mathrm{d}z^{m+1}} \log \Gamma(z),$$ where $\log \Gamma$ denotes the loggamma function. The digamma function $\psi$ is the function $\psi^{(0)}(z)$ and the trigamma function is $\psi^{(1)}(z)$.

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  Domain coloring of $\psi^{(0)}(z)$.

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  Domain coloring of $\psi^{(1)}(z)$.

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  Domain coloring of $\psi^{(2)}(z)$.

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  Domain coloring of $\psi^{(3)}(z)$.

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  Domain coloring of $\psi^{(4)}(z)$.

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  Domain coloring of $\psi^{(5)}(z)$.

Properties

Theorem

The following formula holds for $\mathrm{Re}(z)>0$ and $m>0$: $$\psi^{(m)}(z)=(-1)^{m+1} \displaystyle\int_0^{\infty} \dfrac{t^m e^{-zt}}{1-e^{-t}} \mathrm{d}t,$$ where $\psi^{(m)}$ denotes the polygamma and $e^{-zt}$ denotes the exponential.

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $6.4.1$

See Also

Digamma
Trigamma