 thumb|300px|graphs of the polygamma functions Ï, Ïâ, Ïâ and Ïâ of real arguments in mathematics, the polygamma function of order m is a meromorphic function on \c and defined as the (m+1)-th derivative of the logarithm of the gamma function: :\psi^{(m)}(z) := \frac{d^m}{dz^m} \psi(z) = \frac{d^{m+1}}{dz^{m+1}} \ln\gamma(z). thus :\psi^{(0)}(z) = \psi(z) = \frac{\gamma'(z)}{\gamma(z)} holds where &psi;(z) is the digamma function and &gamma;(z) is the gamma function. they are holomorphic on \c \setminus -\n_0. at all the nonpositive integers these polygamma functions have a pole of order m&nbsp;+&nbsp;1. the function &psi;(1)(z) is sometimes called the trigamma function. {| style="text-align:center" |+ the logarithm of the gamma function and the first few polygamma functions in the complex plane |1000x140px|none |1000x140px|none |1000x140px|none |- | \ln\gamma(z) | \psi^{(0)}(z) | \psi^{(1)}(z) |- |1000x140px|none |1000x140px|none |1000x140px|none |- | \psi^{(2)}(z) | \psi^{(3)}(z) | \psi^{(4)}(z) |} ==integral representation== the polygamma function may be represented as :\begin{align} \psi^{(m)}(z)&= (-1)^{m+1}\int_0^\infty\frac{t^m e^{-zt}} {1-e^{-t}}\ dt\\ &=(-1)^{m}\int_0^1\frac{t^{z-1}}{1-t}\ln^mt\ dt \end{align} which holds for re z &gt;0 and m &gt; 0. for m = 0 see the digamma function definition. ==recurrence relation== it satisfies the recurrence relation :\psi^{(m)}(z+1)= \psi^{(m)}(z) + \frac{(-1)^m\,m!}{z^{m+1}} which â considered for positive integer argument â leads to a presentation of the sum of reciprocals of the powers of the natural numbers: :\frac{\psi^{(m)}(n)}{(-1)^{m+1}\,m!} = \zeta(1+m) - \sum_{k=1}^{n-1} \frac{1}{k^{m+1}} = \sum_{k=n}^\infty \frac{1}{k^{m+1}} \qquad m \ge 1 and :\psi^{(0)}(n) = -\gamma\ + \sum_{k=1}^{n-1}\frac{1}{k} for all n \in \n. like the \ln \gamma-function, the polygamma functions can be generalized from the domain \n uniquely to positive real numbers only due to their recurrence relation and one given function-value, say \psi^{(m)}(1), except in the case m=0 where the additional condition of strictly monotony on \r^+ is still needed. this is a trivial consequence of the bohrâmollerup theorem for the gamma function where strictly logarithmic convexity on \r^+ is demanded additionally. the case m=0 must be treated differently because \psi^{(0)} is not normalizable at infinity (the sum of the reciprocals doesn't converge). ==reflection relation== :(-1)^m \psi^{(m)} (1-z) - \psi^{(m)} (z) = \pi \frac{d^m}{d z^m} \cot{(\pi z)
Gamma_and_related_functions