In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, 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 Mathias Lerch

Mathias Lerch

Mathias Lerch was an eminent Czech mathematician who published about 250 papers, largely on mathematical analysis and number theory...

 http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Lerch.html.

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

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

Root of unity

In mathematics, a root of unity, or de Moivre number, is any complex number that equals 1 when raised to some integer power n. Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, field theory, and the discrete...

, and thus may be expressed as a finite sum over the Hurwitz zeta-function.

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(z)<1/2. Note a general resemblance to a similar series representation for the Hurwitz zeta function.

A Taylor's series in the first parameter was given by Erdélyi

Arthur Erdélyi

Arthur Erdélyi FRS, FRSE was a Hungarian-born British mathematician. Erdélyi was a leading expert on special functions - especially orthogonal polynomials and hypergeometric functions.-Biography:...

. It may be written as the following series, which is valid for

(the correctness of this formula is disputed, please see the talk page)

Please see:
B. R. Johnson,
Generalized Lerch zeta-function.
Pacific J. Math. 53, no. 1 (1974), 189–193.
http://projecteuclid.org/Dienst/UI/1.0/Display/euclid.pjm/1102911791?abstract=

If s is a positive integer, then

A Taylor series

Taylor series

In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....

 in the third variable is given by

Series at a = -n is given by
A special case for n = 0 has the following series

An asymptotic series for
for
and
for

An asymptotic series in the incomplete Gamma function

Incomplete gamma function

In mathematics, the gamma function is defined by a definite integral. The incomplete gamma function is defined as an integral function of the same integrand. There are two varieties of the incomplete gamma function: the upper incomplete gamma function is for the case that the lower limit of...

for

External links

.

• Ramunas Garunkstis, Home Page (2005) (Provides numerous references and preprints.)
• Ramunas Garunkstis, Approximation of the Lerch Zeta Function (PDF)
• S. Kanemitsu, Y. Tanigawa and H. Tsukada, A generalization of Bochner's formula, (undated, 2005 or earlier)