Polygamma functions of negative order. (English) Zbl 0936.33001

Adamchik, Victor S.

J. Comput. Appl. Math. 100, No. 2, 191-199 (1998).

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.

Reviewer: Tom M.Apostol (Pasadena)

MSC:

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

Keywords:

polygamma function; Liouville’s fractional integration; Hurwitz zeta-function; Riemann zeta-function

References:

[1]	Adamchik, V.S., A class of logarithmic integrals, (), 1-8 · Zbl 0922.11114
[2]	Adamchik, V.S.; Srivastava, H.M., Some series of the zeta and related functions, Analysis, 31, 131-144, (1998) · Zbl 0919.11056
[3]	Barnes, E.W., The theory of G-function, Quart. J. math., 31, 264-314, (1899) · JFM 30.0389.02
[4]	Bateman, H.; Erdelyi, A., ()
[5]	Bendersky, L., Sur la function gamma généralisée, Acta math., 61, 263-322, (1933) · Zbl 0008.07001
[6]	Choi, J.; Srivastava, H.M.; Quine, J.R., Some series involving the zeta function, Bull. austral. math. soc., 51, 383-393, (1995) · Zbl 0830.11030
[7]	Finch, S., Glaisher-kinkelin constant, (1996), in HTML essay at URL
[8]	Glaisher, J.W.L., On a numerical continued product, Messenger math., 6, 71-76, (1877)
[9]	Gosper, R.W., \(ʃ\^{}\{m6\}n4\) log \(Г(z) dz\), In special functions, q-series and related topics, Amer. math. soc., 14, (1997)
[10]	Grossman, N., Polygamma functions of arbitrary order, SIAM J. math. anal., 7, 366-372, (1976) · Zbl 0337.33002
[11]	Magnus, W.; Oberhettinger, F.; Soni, R.P., Formulas and theorems for the special functions of mathematical physics, (1966), Springer Berlin · Zbl 0143.08502
[12]	J. Miller, V.S. Adamchik, Derivatives of the Hurwitz Zeta function for rational arguments, J. Comput. Appl. Math., to appear. · Zbl 0928.11037
[13]	Ross, B., Problem 6002, Amer. math. monthly, 81, 1121, (1974)