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Zbl 0936.33001
Adamchik, Victor S.
Polygamma functions of negative order. (English)
[J] J. Comput. Appl. Math. 100, No.2, 191-199 (1998). ISSN 0377-0427

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.
[Tom M.Apostol (Pasadena)]

MSC 2000:: *33B15 Gamma-functions, etc.
11M06 Riemannian zeta-function and Dirichlet L-function

Keywords: polygamma function; Liouville's fractional integration; Hurwitz zeta-function; Riemann zeta-function

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