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.


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