Editing Polygamma function (section)

==Series representation==
The polygamma function has the series representation

:<math>\psi^{(m)}(z) = (-1)^{m+1}\, m! \sum_{k=0}^\infty \frac{1}{(z+k)^{m+1}}</math>

which holds for integer values of {{math|''m'' > 0}} and any complex {{mvar|z}} not equal to a negative integer.  This representation can be written more compactly in terms of the [[Hurwitz zeta function]] as

:<math>\psi^{(m)}(z) = (-1)^{m+1}\, m!\, \zeta (m+1,z).</math>

This relation can for example be used to compute the special values<ref>
{{cite journal|first1=K. S. |last1=Kölbig|year=1996|journal=Journal of Computational and Applied Mathematics |volume=75|number=1|pages=43–46|title=The polygamma function <math>\psi^{(k)}(x)</math> for <math>x=\frac{1}{4}</math> and <math>x=\frac{3}{4}</math>|doi=10.1016/S0377-0427(96)00055-6|doi-access=free}}
</ref>
:<math>
\psi^{(2n-1)}\left(\frac14\right) = \frac{4^{2n-1}}{2n}\left(\pi^{2n}(2^{2n}-1)|B_{2n}|+2(2n)!\beta(2n)\right);
</math>
:<math>
\psi^{(2n-1)}\left(\frac34\right) = \frac{4^{2n-1}}{2n}\left(\pi^{2n}(2^{2n}-1)|B_{2n}|-2(2n)!\beta(2n)\right);
</math>
:<math>
\psi^{(2n)}\left(\frac14\right) = -2^{2n-1}\left(\pi^{2n+1}|E_{2n}|+2(2n)!(2^{2n+1}-1)\zeta(2n+1)\right);
</math>
:<math>
\psi^{(2n)}\left(\frac34\right) = 2^{2n-1}\left(\pi^{2n+1}|E_{2n}|-2(2n)!(2^{2n+1}-1)\zeta(2n+1)\right).
</math>

Alternately, the Hurwitz zeta can be understood to generalize the polygamma to arbitrary, non-integer order.

One more series may be permitted for the polygamma functions. As given by [[Schlömilch]],

:<math>\frac{1}{\Gamma(z)} = z e^{\gamma z}  \prod_{n=1}^\infty \left(1 + \frac{z}{n}\right) e^{-\frac{z}{n}}.</math>

This is a result of the [[Weierstrass factorization theorem]]. Thus, the gamma function may now be defined as:

:<math>\Gamma(z) = \frac{e^{-\gamma z}}{z} \prod_{n=1}^\infty \left(1 + \frac{z}{n}\right)^{-1} e^\frac{z}{n}.</math>

Now, the [[natural logarithm]] of the gamma function is easily representable:

:<math>\ln \Gamma(z) = -\gamma z - \ln(z) + \sum_{k=1}^\infty \left( \frac{z}{k} - \ln\left(1 + \frac{z}{k}\right) \right).</math>

Finally, we arrive at a summation representation for the polygamma function:

:<math>\psi^{(n)}(z) = \frac{\mathrm{d}^{n+1}}{\mathrm{d}z^{n+1}}\ln \Gamma(z) = -\gamma \delta_{n0} - \frac{(-1)^n n!}{z^{n+1}} + \sum_{k=1}^{\infty} \left(\frac{1}{k} \delta_{n0} - \frac{(-1)^n n!}{(k+z)^{n+1}}\right)</math>

Where {{math|''δ''<sub>''n''0</sub>}} is the [[Kronecker delta]].

Also the [[Lerch transcendent]]
 
:<math>\Phi(-1, m+1, z) = \sum_{k=0}^\infty \frac{(-1)^k}{(z+k)^{m+1}}</math>

can be denoted in terms of polygamma function

:<math>\Phi(-1, m+1, z)=\frac1{(-2)^{m+1}m!}\left(\psi^{(m)}\left(\frac{z}{2}\right)-\psi^{(m)}\left(\frac{z+1}{2}\right)\right)</math>