Lerch zeta function

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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].


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


A contour integral representation is given by


where the contour must not enclose any of the points

A Hermite-like integral representation is given by




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


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:



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 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.[permanent dead link]

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


An asymptotic series in the incomplete gamma function



The Lerch transcendent is implemented as LerchPhi in Maple.


External links[edit]