Lerch zeta function


In mathematics, the Lerch zeta function, sometimes called the Hurwitz-Lerch zeta-function, is a special function that generalizes the Hurwitz zeta function and the polylogarithm. It is named after the Czech mathematician Mathias Lerch .

Definition

The Lerch zeta function is given by
A related function, the Lerch transcendent, is given by
The two are related, as

Integral representations

An integral representation is given by
for
A contour integral representation is given by
for
where the contour must not enclose any of the points
A Hermite-like integral representation is given by
for
and
for
Similar representations include
and
holding for positive z. Furthermore,
The last formula is also known as Lipschitz formula.

Special cases

The Hurwitz zeta function is a special case, given by
The polylogarithm is a special case of the Lerch Zeta, given by
The Legendre chi function is a special case, given by
The Riemann zeta function is given by
The Dirichlet eta function is given by

Identities

For λ rational, the summand is a root of unity, and thus may be expressed as a finite sum over the Hurwitz zeta-function. Suppose with and. Then and.
Various identities include:
and
and

Series representations

A series representation for the Lerch transcendent is given by
The series is valid for all s, and for complex z with Re<1/2. Note a general resemblance to a similar series representation for the Hurwitz zeta function.
A Taylor series in the first parameter was given by Erdélyi. It may be written as the following series, which is valid for
If n is a positive integer, then
where is the digamma function.
A Taylor series in the third variable is given by
where is the Pochhammer symbol.
Series at a = -n is given by
A special case for n = 0 has the following series
where is the polylogarithm.
An asymptotic series for
for
and
for
An asymptotic series in the incomplete gamma function
for

Asymptotic expansion

The polylogarithm function is defined as
Let
For and, an asymptotic expansion of
for large and fixed and is given by
for, where is the Pochhammer symbol.
Let
Let
be its Taylor coefficients at. Then for fixed and
as.

Software

The Lerch transcendent is implemented as .