Lerch zeta function

From Wikipedia, the free encyclopedia
  (Redirected from Lerch transcendent)
Jump to: navigation, search

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 [1].

Definition[edit]

The Lerch zeta-function is given by

A related function, the Lerch transcendent, is given by

The two are related, as

Integral representations[edit]

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[edit]

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[edit]

For λ rational, the summand is a root of unity, and thus may be expressed as a finite sum over the Hurwitz zeta-function.

Various identities include:

and

and

Series representations[edit]

A series representation for the Lerch transcendent is given by

(Note that is a binomial coefficient.)

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

B. R. Johnson (1974). "Generalized Lerch zeta-function". Pacific J. Math. 53 (1): 189–193. 

If s 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

Software[edit]

The Lerch transcendent is implemented as LerchPhi in Maple.

References[edit]

External links[edit]

  • "§25.14, Lerch's Transcendent". NIST Digital Library of Mathematical Functions. National Institute of Standards and Technology. 2010. Retrieved 28 January 2012.