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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 [1].

Contents

DefinitionEdit

The Lerch zeta function is given by

 

A related function, the Lerch transcendent, is given by

 

The two are related, as

 

Integral representationsEdit

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 casesEdit

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

 

IdentitiesEdit

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 representationsEdit

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

 

for  

An asymptotic series in the incomplete gamma function

 

for  

SoftwareEdit

The Lerch transcendent is implemented as LerchPhi in Maple.

ReferencesEdit

External linksEdit