The Lerch zeta function (after Mathias Lerch ) is a very general zeta function . Very many series of reciprocal powers (including the Hurwitz zeta function and the polylogarithm ) can be represented as a special case of this function.
definition
The two functions
and
are called Lerch's zeta function . The relationship between the two is through
given.
Special cases and special values
The following special cases also apply (selection):
Furthermore is
with the Catalan constant , the Glaisher-Kinkelin constant and the Apéry constant of the Riemann zeta function.
More formulas
Integral representations
One possible integral representation is
-
For
The curve integral
with must not contain the points .
Furthermore is
for and .
Likewise is
for .
Series representations
A series representation for the transcendent lark is
It applies to all and complex with ; compare the series representation of the Hurwitz zeta function .
If is positive and whole, then applies
A Taylor series of the third variable is through
given for using the Pochhammer symbol .
The following applies in the
limit value
-
.
The special case has the following series:
for .
The asymptotic expansion for is given by
for and
if .
Using the incomplete gamma function holds
with and .
Identities and other formulas
Furthermore applies to the integral representation with or
and
-
.
literature
-
Mathias Lerch : Démonstration élémentaire de la formule :, L'Enseignement Mathématique 5 (1903): pp. 450–453
- M. Jackson: On Lerch's transcendent and the basic bilateral hypergeometric series , J. London Math. Soc. 25 (3), 1950: pp. 189-196
- Jesús Guillera, Jonathan Sondow: Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent . In: Ramanujan J. Volume 16, Number 3, 2008, pages 247-270; see. in arxiv
- Antanas Laurinčikas and Ramūnas Garunkštis: The Lerch zeta-function , Dordrecht: Kluwer Academic Publishers, 2002, ISBN 978-1-4020-1014-9 online
Web links
Individual evidence
-
↑ http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/LerchPhi/03/ShowAll.html
-
↑ Guillera, Sondow 2008, Theorem 3.1 (see Ref.)