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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:
 [1] Adamchik, V. S.: A class of logarithmic integrals. Proc. ISSAC’97, 1-8 (1997) · 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) · Zbl 30.0389.02 [4] Bateman, H.; Erdelyi, A.: 2nd ed. Higher transcendental functions. Higher transcendental functions 1 (1953) [5] Bendersky, L.: Sur la function gamma généralisée. Acta math. 61, 263-322 (1933) · Zbl 59.0373.02 [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) [8] Glaisher, J. W. L.: On a numerical continued product. Messenger math. 6, 71-76 (1877) [9] Gosper, R. W.: ? m 6 n 4 $\log {\cyr G}(z)$dz. Amer. math. Soc. 14 (1997) · Zbl 0870.33001 [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) · 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)