zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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:
33B15Gamma, beta and polygamma functions
11M06$\zeta (s)$ and $L(s, \chi)$
WorldCat.org
Full Text: DOI
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)