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 $\psi^{(1)}(z)$.

• 

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

• 

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

• 

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

• 

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

• 

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

• 

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

Properties

Integral representation of polygamma for Re(z) greater than 0
Integral representation of polygamma 2
Polygamma recurrence relation
Polygamma reflection formula
Polygamma multiplication formula
Polygamma series representation
Value of polygamma at 1
Value of polygamma at positive integer
Value of polygamma at 1/2
Value of derivative of trigamma at positive integer plus 1/2
Relation between polygamma and Hurwitz zeta
Series for polygamma in terms of Riemann zeta

See Also

Digamma
Trigamma

References

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