Lerch zeta function
The Lerchsche zeta function ( after Mathias Lerch ) is a very general zeta function. Very many series of reciprocal powers ( including the Hurwitz zeta function and the Polylogarithmus ) can be represented as a special case of this function.
Definition
The two functions
And
Are referred to as Lerchsche zeta function. The relationship between the two is by
Given.
Special cases and special values
- The Hurwitz zeta function:
- The Polylogarithmus:
- The Legendre chi function:
- The Riemann ζ - function:
- The Dirichlet η - function:
- The Dirichlet beta function:
In addition, the following special cases (Selection) shall apply:
Furthermore, it is
With the catalanschen constants, the Glaisher - Kinkelin constants and the apery constants of the Riemann zeta function.
Other formulas
Integral representations
A possible integral representation is
The line integral
Must not contain periods.
Furthermore, it is
And
Series representations
A series representation for the transcendental Lerch
It applies to all and complex with; they compare to the series representation of the Hurwitz zeta function.
If is positive and quite, shall
A Taylor series of the third variable is by
Given.
Is is true
The special case has the following series:
The asymptotic expansion for is given by
And
Applies Using the incomplete gamma function