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 $$L(s, \chi)$$
Full Text:

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)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.