## Polygamma functions of negative order.(English)Zbl 0936.33001

For positive integer $$n$$ the polygamma function $$\psi^{(n)} (z)$$ is defined to be the derivative of order $$n+1$$ of $$\log \Gamma(z)$$. The definition can be extended to negative integer $$n$$ by Liouville’s fractional integration, which gives $\psi^{-n)} (z)= {1\over(n-1)!} \int^z_0 (z-t)^{n-2} \log \Gamma(t) dt.$ The author replaces $$\log\Gamma(t)$$ by a series representation and integrates term by term to express $$n!\psi^{(-n)}(z)$$ as an explicit polynomial in $$z$$ plus a term $$n\zeta'(1-n,z)$$ where $$R(z)>0$$ and $$\zeta'(1-n,z)$$ is the derivative with respect to $$s$$ of the Hurwitz zeta-function $$\zeta(s,z)$$ evaluated at $$s=1-n$$. For example, $\psi^{(-2)}(z)= \textstyle {1\over 2}z(1-z)+ \textstyle {1\over 2} z\log(2\pi)-\zeta'(-1)+ \zeta'(-1,z),$ where $$\zeta(s)=\zeta(s,1)$$ is the Riemann zeta-function.

### MSC:

 33B15 Gamma, beta and polygamma functions 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$
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### References:

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